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4 Integração 4.2 Propriedades de integração 4.4 Integração por substituição

4.3 Regras Básicas de Integração

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A integral indefinida de uma função $f$ em relação a $x$ é

\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}% \pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int f(x)\,dx=% F(x)+C,

(4.116)

onde $F$ é uma primitiva de $f$ , i.e. $F^{\prime}=f$ , e $C$ é uma constante indeterminada. Na sequência, vamos estudar as regras básicas para o cálculo de integrais.

4.3.1 Integral de Função Potência

Com base na derivada de função potência, podemos afirmar que

\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}% \pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int x^{r}\,dx% =\frac{x^{r+1}}{r+1}+C,\leavevmode\nobreak\ r\neq-1.

(4.117)

De fato, para $r\neq-1$ , temos

$\displaystyle F(x)$	$\displaystyle=\frac{x^{r+1}}{r+1}+C$	(4.118)
$\displaystyle F^{\prime}(x)$	$\displaystyle=(r+1)\frac{x^{r}}{r+1}$	(4.119)
	$\displaystyle=x^{r}.$	(4.120)

Exemplo 4.3.1.

Estudamos os seguintes casos:

a)

$\int x\,dx=\frac{x^{2}}{2}+C.$ (4.121)

Código 71: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(x, x)

4print(I)
```
x**2/2
```
b)

$\displaystyle\int\frac{1}{x^{2}}\,dx$ $\displaystyle=\int x^{-2}\,dx$ (4.122)

$\displaystyle=-x^{-1}+C$ (4.123)

$\displaystyle=-\frac{1}{x}+C.$ (4.124)

Código 72: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(1/x**2, x)

4print(I)
```
-1/x
```

Exemplo 4.3.2.

Vamos calcular

\int_{-1}^{1}x^{2}\,dx.

(4.125)

Da regra da potência (4.117), temos

\int x^{2}\,dx=\frac{x^{3}}{3}+C.

(4.126)

Logo, do teorema fundamental do cálculo, temos

$\displaystyle\int_{-1}^{1}x^{2}\,dx$	$\displaystyle=\left.\frac{x^{3}}{3}\right\|_{-1}^{1}$	(4.127)
	$\displaystyle=\frac{1^{3}}{3}-\frac{(-1)^{3}}{3}$	(4.128)
	$\displaystyle=\frac{1}{3}+\frac{1}{3}=\frac{2}{3}.$	(4.129)

Código 73: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(x**2, (x, -1, 1))

4print(I)

2/3

4.3.2 Regra da Multiplicação por Constante

Seja $k$ uma constante. Então, temos a seguinte regra da multiplicação por constante

\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}% \pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int k\cdot f(% x)\,dx=k\cdot\int f(x)\,dx

(4.130)

De fato, se $F$ é uma primitiva de $f$ , então pela regra da multiplicação por constante para derivadas, temos

	$\displaystyle(k\cdot F)^{\prime}$	$\displaystyle=k\cdot F^{\prime}$		(4.131)
		$\displaystyle=k\cdot f,$		(4.132)

i.e. $k\cdot F$ é primitiva de $k\cdot f$ .

Exemplo 4.3.3.

Estudamos os seguintes casos:

a)

$\displaystyle\int 2x\,dx$ $\displaystyle=2\int x\,dx$ (4.133)

$\displaystyle=2\left(\frac{x^{2}}{2}+C\right)$ (4.134)

$\displaystyle=x^{2}+2C$ (4.135)

$\displaystyle=x^{2}+C$ (4.136)

Aqui, fizemos um abuso de linguagem ao assumir $2C=C$ . Isso pode ser feito, pois $C$ denota uma constante indeterminada e, multiplicá-la por dois continua sendo indeterminada e constante. Vamos fazer este tipo de simplificação de notação várias vezes ao longo do texto.

Código 74: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(2*x, x)

4print(I)
```
x**2
```
b)

$\displaystyle\int\frac{1}{3}\sqrt{x}\,dx$ $\displaystyle=\frac{1}{3}\int x^{\frac{1}{2}}\,dx$ (4.137)

$\displaystyle=\frac{1}{3}\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C$ (4.138)

$\displaystyle=\frac{1}{3}\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+C$ (4.139)

$\displaystyle=\frac{2}{9}\sqrt{x^{3}}+C$ (4.140)

Código 75: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(1/3*sp.sqrt(x), x)

4print(I)
```
0.222222222222222*x**(3/2)
```
c)

$\displaystyle\int_{0}^{1}-x^{2}\,dx$ $\displaystyle=-\int_{0}^{1}x^{2}\,dx$ (4.141)

$\displaystyle=-\left[\frac{x^{3}}{3}\right]_{0}^{1}$ (4.142)

$\displaystyle=-\left(\frac{1}{3}-\frac{0}{3}\right)$ (4.143)

$\displaystyle=-\frac{1}{3}.$ (4.144)

Código 76: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(-x**2, (x, 0, 1))

4print(I)
```
-1/3
```

4.3.3 Regra da soma ou subtração

Se $f$ e $g$ são funções integráveis, então vale a seguinte regra da soma/subtração

\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}% \pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int\left[f(x)% \pm g(x)\right]\,dx=\int f(x)\,dx\pm\int g(x)\,dx.

(4.145)

De fato, sejam $F$ uma primitiva de $f$ e $G$ uma primitiva de $g$ . Temos

	$\displaystyle(F\pm G)^{\prime}$	$\displaystyle=F^{\prime}\pm G^{\prime}$		(4.146)
		$\displaystyle=f\pm g,$		(4.147)

i.e. $F\pm G$ é primitiva de $f\pm g$ .

Exemplo 4.3.4.

Estudamos os seguintes casos:

a)

$\displaystyle\int{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}x}+{% \color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}% \pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}1}\,dx$ $\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int x\,% dx}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}% \pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}\int dx}$ (4.148)

$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\frac{x^% {2}}{2}+C_{1}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{% 1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}x+C_{2}}$ (4.149)

$\displaystyle=\frac{x^{2}}{2}+x+C$ (4.150)

Aqui, $C_{1}$ , $C_{2}$ e $C=C_{1}+C_{2}$ denotam constantes indeterminadas.

Código 77: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(x+1, x)

4print(I)
```
x**2/2 + x
```
b)

$\displaystyle\int{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\sqrt{x}% }-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}% \pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}x}\,dx$ $\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int x^{% \frac{1}{2}}\,dx}-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{% 1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}\int x\,dx}$ (4.151)

$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\frac{2}% {3}x^{\frac{3}{2}}}-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb% }{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}\frac{% x^{2}}{2}}+C$ (4.152)

Código 78: Python

⬇

1import sympy as sp

2from sympy.abc import x

3f = sp.sqrt(x) - x

4I = sp.integrate(f, x)

5print(I)
```
2*x**(3/2)/3 - x**2/2
```

$\displaystyle\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}% \pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int(2x^{2}+3x% -1)\,dx$	$\displaystyle=\int\left[2x^{2}+(3x-1)\right]\,dx$	(4.153)
	$\displaystyle=\int 2x^{2}\,dx+\int 3x-1\,dx$	(4.154)
	$\displaystyle=\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1% }\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int 2x^{2}\,% dx-\int 3x\,dx-\int\,dx$	(4.155)
	$\displaystyle=2\int x^{2}\,dx+3\int x\,dx-\int\,dx$	(4.156)
	$\displaystyle=\frac{2}{3}x^{3}+\frac{3}{2}x^{2}-x+C$	(4.157)

Código 79: Python

⬇

1import sympy as sp

2from sympy.abc import x

3f = 2*x**2 + 3*x - 1

4I = sp.integrate(f, x)

5print(I)

2*x**3/3 + 3*x**2/2 - x

Exemplo 4.3.5.

Vamos calcular

\int_{0}^{1}x^{2}+1\,dx.

(4.158)

Temos

	$\displaystyle\int x^{2}+1\,dx$	$\displaystyle=\int x^{2}\,dx+\int\,dx$		(4.159)
		$\displaystyle=\frac{x^{3}}{3}+x+C.$		(4.160)

Agora, do teorema fundamental do cálculo, temos

$\displaystyle\int_{0}^{1}x^{2}+1\,dx$	$\displaystyle=\left.\frac{x^{3}}{3}+x\right\|_{0}^{1}$	(4.161)
	$\displaystyle=\left(\frac{1}{3}+1\right)-\left(\frac{0^{3}}{3}+0\right)$	(4.162)
	$\displaystyle=\frac{4}{3}.$	(4.163)

Código 80: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(x**2 + 1, (x, 0, 1))

4print(I)

4/3

4.3.4 Integral de $x^{-1}$

Começamos lembrando que

\frac{d}{dx}\ln x=\frac{1}{x},\leavevmode\nobreak\ x>0.

(4.164)

Para $x<0$ , usamos a regra da cadeia

$\displaystyle\frac{d}{dx}\ln(-x)$	$\displaystyle=\frac{1}{-x}\cdot(-x)^{\prime}$	(4.165)
	$\displaystyle=-\frac{1}{x}\cdot(-1)$	(4.166)
	$\displaystyle=\frac{1}{x}$	(4.167)

Ou seja, temos que

\frac{d}{dx}\ln|x|=\frac{1}{x},

(4.168)

donde concluímos que a integral de $x^{-1}$ é

\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}% \pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int\frac{1}{x% }\,dx=\ln|x|+C.

(4.169)

Exemplo 4.3.6.

$\displaystyle\int_{1}^{e}x^{-1}\,dx$	$\displaystyle=\int_{1}^{e}\frac{1}{x}\,dx$	(4.170)
	$\displaystyle=\left[\ln\|x\|\right]_{1}^{e}$	(4.171)
	$\displaystyle=\ln\|e\|-\ln\|1\|$	(4.172)
	$\displaystyle=1-0=1.$	(4.173)

Código 81: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(1/x, (x, 1, sp.E))

4print(I)

4.3.5 Integral da Função Exponencial Natural

Da derivada da função exponencial natural, temos

\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}% \pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int e^{x}\,dx% =e^{x}+C.

(4.174)

Exemplo 4.3.7.

Vamos estudar os seguintes casos:

a)

$\displaystyle\int e^{2+x}\,dx$ $\displaystyle=\int e^{2}e^{x}\,dx$ (4.175)

$\displaystyle=e^{2}\int e^{x}\,dx$ (4.176)

$\displaystyle=e^{2}e^{x}+C$ (4.177)

$\displaystyle=e^{2+x}+C$ (4.178)

Código 82: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(sp.exp(2+x), x)

4print(I)
```
exp(x + 2)
```
b)

$\displaystyle\int_{0}^{\ln 2}e^{x}\,dx$ $\displaystyle=\left.e^{x}\right|_{0}^{\ln 2}$ (4.179)

$\displaystyle=e^{\ln 2}-e^{0}$ (4.180)

$\displaystyle=2-1$ (4.181)

$\displaystyle=1$ (4.182)

Código 83: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(sp.exp(x), (x, 0, sp.log(2)))

4print(I)
```
1
```

4.3.6 Integrais de Funções Trigonométricas

No que lembramos que

\frac{d}{dx}\cos(x)=-\operatorname{sen}(x)

(4.183)

temos que a integral da função seno é

\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}% \pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int% \operatorname{sen}(x)\,dx=-\cos(x)+C.

(4.184)

Exemplo 4.3.8.

Estudamos os seguintes casos:

a)

$\displaystyle\int 2\operatorname{sen}(x)\,dx$ $\displaystyle=2\int\operatorname{sen}(x)dx$ (4.185)

$\displaystyle=-2\cos(x)+C$ (4.186)

Código 84: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(2*sp.sin(x), x)

4print(I)
```
-2*cos(x)
```
b)

$\displaystyle\int_{-\pi}^{\pi}\operatorname{sen}(x)\,dx$ $\displaystyle=\left.-\cos(x)\right|_{-\pi}^{\pi}$ (4.187)

$\displaystyle=-\cos(\pi)-[-\cos(-\pi)]$ (4.188)

$\displaystyle=1-1=0$ (4.189)

Código 85: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(sp.sin(x), (x, -sp.pi, sp.pi))

4print(I)
```
0
```

Também, lembramos que

\frac{d}{dx}\operatorname{sen}(x)=\cos(x),

(4.190)

donde temos que a integral da função cosseno

\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}% \pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int\cos(x)\,% dx=\operatorname{sen}(x)+C.

(4.191)

Exemplo 4.3.9.

Estudamos os seguintes casos:

a)

$\displaystyle\int\frac{1}{2}\cos(x)\,dx$ $\displaystyle=\frac{1}{2}\int\cos(x)\,dx$ (4.192)

$\displaystyle=\frac{1}{2}\operatorname{sen}(x)+C$ (4.193)

Código 86: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(1/2*sp.cos(x), x)

4print(I)
```
0.5*sin(x)
```
b)

$\displaystyle\int_{-\pi}^{\pi}\cos(x)\,dx$ $\displaystyle=\left.\operatorname{sen}(x)\right|_{-\pi}^{\pi}$ (4.195)

$\displaystyle=\operatorname{sen}(\pi)-\operatorname{sen}(-\pi)$ (4.196)

$\displaystyle=0$ (4.197)

Código 87: Python

⬇

1import sympy as sp

2from sympy.abc import x

3I = sp.integrate(sp.cos(x), (x, -sp.pi, sp.pi))

4print(I)
```
0
```

4.3.7 Tabela de Integrais

	$\displaystyle\int k\cdot f(x)\,dx=k\cdot\int f(x)\,dx$		(4.198)
	$\displaystyle\int\left[f(x)\pm g(x)\right]\,dx=\int f(x)\,dx\pm\int g(x)\,dx$		(4.199)
	$\displaystyle\int x^{r}\,dx=\frac{x^{r+1}}{r+1}+C,\quad r\neq-1$		(4.200)
	$\displaystyle\int\frac{1}{x}\,dx=\ln x+C$		(4.201)
	$\displaystyle\int e^{x}\,dx=e^{x}+C$		(4.202)
	$\displaystyle\int\operatorname{sen}(x)\,dx=-\cos(x)+C$		(4.203)
	$\displaystyle\int\cos(x)\,dx=\operatorname{sen}(x)+C$		(4.204)

4.3.8 Exercícios resolvidos

ER 4.3.1.

Calcule

\int\frac{x^{2}+2x}{\sqrt{x}}\,dx

(4.205)

Solução.

$\displaystyle\int\frac{x^{2}+2x}{\sqrt{x}}\,dx$	$\displaystyle=\int\frac{x^{2}}{\sqrt{x}}+2\frac{x}{\sqrt{x}}\,dx$	(4.206)
	$\displaystyle=\int\frac{x^{2}}{x^{\frac{1}{2}}}+2\frac{x}{x^{\frac{1}{2}}}\,dx$	(4.207)
	$\displaystyle=\int x^{2-\frac{1}{2}}+2x^{1-\frac{1}{2}}\,dx$	(4.208)
	$\displaystyle=\int x^{\frac{3}{2}}\,dx+2\int x^{\frac{1}{2}}\,dx$	(4.209)

Agora, usando a regra da função potência (4.117), obtemos

	$\displaystyle\int\frac{x^{2}+2x}{\sqrt{x}}\,dx$	$\displaystyle=\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+2\frac{x^{\frac{1}{2}+1}% }{\frac{1}{2}+1}+C$		(4.210)
		$\displaystyle=\frac{2}{5}x^{\frac{5}{2}}+\frac{4}{3}x^{\frac{3}{2}}+C$		(4.211)

Com o Python+SymPy, podemos computar a solução deste exercício

Código 88: Python

⬇

1 >>> from sympy import *

2 >>> x = symbols('x')

3 >>> integrate((x**2+2*x)/(sqrt(x)))

4 2*x**(5/2)/5 + 4*x**(3/2)/3

ER 4.3.2.

Calcule

\int_{1}^{e}\frac{1}{2x}\,dx.

(4.212)

Solução.

Das regras básicas de integração, temos

$\displaystyle\int\frac{1}{2x}\,dx$	$\displaystyle=\int\frac{1}{2}\cdot\frac{1}{x}\,dx$	(4.213)
	$\displaystyle=\frac{1}{2}\int\frac{1}{x}\,dx$	(4.214)
	$\displaystyle=\frac{1}{2}\ln(x)+C$	(4.215)
	$\displaystyle=\ln\sqrt{x}+C.$	(4.216)

Então, do Teorema fundamental do cálculo, temos

$\displaystyle\int_{1}^{e}\frac{1}{2x}\,dx$	$\displaystyle=\left.\ln\sqrt{x}\right\|_{1}^{e}$	(4.217)
	$\displaystyle=\ln(\sqrt{e})-\ln(1)$	(4.218)
	$\displaystyle=\frac{1}{2}$	(4.219)

Com o Python+SymPy, computamos a solução como segue

Código 89: Python

⬇

1 >>> from sympy import *

2 >>> x = symbols('x')

3 >>> integrate(1/(2*x), (x, 1, E))

4 1/2

4.3.9 Exercícios

E. 4.3.1.

Calcule

a)

$\displaystyle\int\,dx$
b)

$\displaystyle\int x^{-2}\,dx$
c)

$\displaystyle\int\sqrt{x}\,dx$
d)

$\displaystyle\int\frac{1}{\sqrt{x}}\,dx$

Resposta.

a) $\displaystyle x+C$ ; b) $\displaystyle-\frac{1}{x}+C$ ; c) $\displaystyle\frac{2}{3}x^{3/2}+C$ ; d) $\displaystyle 2x^{1/2}+C$

E. 4.3.2.

Calcule

a)

$\displaystyle\int 1+x^{-2}\,dx$
b)

$\displaystyle\int x-\frac{1}{x}\,dx$
c)

$\displaystyle\int 2x^{3}-3x^{2}+1\,dx$

Resposta.

a) $\displaystyle x-\frac{1}{x}+C$ ; b) $\displaystyle\frac{x^{2}}{2}-\ln|x|+C$ ; c) $\displaystyle\frac{1}{2}x^{4}-x^{3}+x+C$

E. 4.3.3.

Calcule

a)

$\displaystyle\int 2\cos(x)\,dx$
b)

$\displaystyle\int 1-\operatorname{sen}(x)\,dx$

Resposta.

a) $\displaystyle 2\operatorname{sen}(x)+C$ ; b) $\displaystyle x+\cos(x)+C$

E. 4.3.4.

Calcule

a)

$\displaystyle\int_{-1}^{1}x^{3}\,dx$
b)

$\displaystyle\int_{e}^{2e}x^{-1}\,dx$

Resposta.

a) $0$ ; b) $\ln(2)$ ;

E. 4.3.5.

Calcule

1.

$\displaystyle\int_{0}^{1}x^{2}-2x^{3}\,dx$
2.

$\displaystyle\int_{1}^{2}\frac{x+1}{\sqrt{x}}\,dx$

Resposta.

a) $\displaystyle-\frac{1}{6}$ ; b) $\displaystyle\frac{10}{3}\sqrt{2}-\frac{8}{3}$ ;

E. 4.3.6.

Cálculo

a)

$\displaystyle\int_{0}^{\frac{\pi}{2}}\operatorname{sen}(x)\,dx$
b)

$\displaystyle\int_{0}^{\frac{\pi}{2}}\cos(x)\,dx$
c)

$\displaystyle\int_{0}^{\pi}\cos(x)-\operatorname{sen}(x)\,dx$

Resposta.

a) $1$ ; b) $1$ ; c) $-2$

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| | | |

$\displaystyle\int\frac{1}{x^{2}}\,dx$	$\displaystyle=\int x^{-2}\,dx$	(4.122)
	$\displaystyle=-x^{-1}+C$	(4.123)
	$\displaystyle=-\frac{1}{x}+C.$	(4.124)

$\displaystyle\int 2x\,dx$	$\displaystyle=2\int x\,dx$	(4.133)
	$\displaystyle=2\left(\frac{x^{2}}{2}+C\right)$	(4.134)
	$\displaystyle=x^{2}+2C$	(4.135)
	$\displaystyle=x^{2}+C$	(4.136)

$\displaystyle\int\frac{1}{3}\sqrt{x}\,dx$	$\displaystyle=\frac{1}{3}\int x^{\frac{1}{2}}\,dx$	(4.137)
	$\displaystyle=\frac{1}{3}\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C$	(4.138)
	$\displaystyle=\frac{1}{3}\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+C$	(4.139)
	$\displaystyle=\frac{2}{9}\sqrt{x^{3}}+C$	(4.140)

$\displaystyle\int_{0}^{1}-x^{2}\,dx$	$\displaystyle=-\int_{0}^{1}x^{2}\,dx$	(4.141)
	$\displaystyle=-\left[\frac{x^{3}}{3}\right]_{0}^{1}$	(4.142)
	$\displaystyle=-\left(\frac{1}{3}-\frac{0}{3}\right)$	(4.143)
	$\displaystyle=-\frac{1}{3}.$	(4.144)

$\displaystyle\int{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}x}+{% \color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}% \pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}1}\,dx$	$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int x\,% dx}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}% \pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}\int dx}$	(4.148)
	$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\frac{x^% {2}}{2}+C_{1}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{% 1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}x+C_{2}}$	(4.149)
	$\displaystyle=\frac{x^{2}}{2}+x+C$	(4.150)

	$\displaystyle\int{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\sqrt{x}% }-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}% \pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}x}\,dx$	$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\int x^{% \frac{1}{2}}\,dx}-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{% 1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}\int x\,dx}$		(4.151)
		$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{% 0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@color@rgb@fill{0}{0}{1}\frac{2}% {3}x^{\frac{3}{2}}}-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb% }{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@color@rgb@fill{1}{0}{0}\frac{% x^{2}}{2}}+C$		(4.152)

$\displaystyle\int e^{2+x}\,dx$	$\displaystyle=\int e^{2}e^{x}\,dx$	(4.175)
	$\displaystyle=e^{2}\int e^{x}\,dx$	(4.176)
	$\displaystyle=e^{2}e^{x}+C$	(4.177)
	$\displaystyle=e^{2+x}+C$	(4.178)

$\displaystyle\int_{0}^{\ln 2}e^{x}\,dx$	$\displaystyle=\left.e^{x}\right\|_{0}^{\ln 2}$	(4.179)
	$\displaystyle=e^{\ln 2}-e^{0}$	(4.180)
	$\displaystyle=2-1$	(4.181)
	$\displaystyle=1$	(4.182)

	$\displaystyle\int 2\operatorname{sen}(x)\,dx$	$\displaystyle=2\int\operatorname{sen}(x)dx$		(4.185)
		$\displaystyle=-2\cos(x)+C$		(4.186)

$\displaystyle\int_{-\pi}^{\pi}\operatorname{sen}(x)\,dx$	$\displaystyle=\left.-\cos(x)\right\|_{-\pi}^{\pi}$	(4.187)
	$\displaystyle=-\cos(\pi)-[-\cos(-\pi)]$	(4.188)
	$\displaystyle=1-1=0$	(4.189)

	$\displaystyle\int\frac{1}{2}\cos(x)\,dx$	$\displaystyle=\frac{1}{2}\int\cos(x)\,dx$		(4.192)
		$\displaystyle=\frac{1}{2}\operatorname{sen}(x)+C$		(4.193)

$\displaystyle\int_{-\pi}^{\pi}\cos(x)\,dx$	$\displaystyle=\left.\operatorname{sen}(x)\right\|_{-\pi}^{\pi}$	(4.195)
	$\displaystyle=\operatorname{sen}(\pi)-\operatorname{sen}(-\pi)$	(4.196)
	$\displaystyle=0$	(4.197)

4.3 Regras Básicas de Integração

4.3.1 Integral de Função Potência

Exemplo 4.3.1.

Exemplo 4.3.2.

4.3.2 Regra da Multiplicação por Constante

Exemplo 4.3.3.

4.3.3 Regra da soma ou subtração

Exemplo 4.3.4.

Exemplo 4.3.5.

4.3.4 Integral de x−1

Exemplo 4.3.6.

4.3.5 Integral da Função Exponencial Natural

Exemplo 4.3.7.

4.3.6 Integrais de Funções Trigonométricas

Exemplo 4.3.8.

Exemplo 4.3.9.

4.3.7 Tabela de Integrais

4.3.8 Exercícios resolvidos

ER 4.3.1.

Solução.

ER 4.3.2.

Solução.

4.3.9 Exercícios

E. 4.3.1.

Resposta.

E. 4.3.2.

Resposta.

E. 4.3.3.

Resposta.

E. 4.3.4.

Resposta.

E. 4.3.5.

Resposta.

E. 4.3.6.

Resposta.

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4.3.4 Integral de $x^{-1}$