Question

Calcular `int_0^4sen(5x)cos^2(5x)dx`

Answer 1

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Calcular a integral definida:

$\mathsf{\displaystyle\int_0^4 sen(5x)\,cos^2(5x)\,dx}\\\\\\ =\mathsf{\displaystyle\int_0^4 cos^2(5x)\cdot sen(5x)\,dx}\\\\\\ =\mathsf{\displaystyle\int_0^4 (-5)\,cos^2(5x)\cdot sen(5x)\,dx}\\\\\\ =\mathsf{\displaystyle-\,\frac{1}{5}\int_0^4 cos^2(5x)\cdot (-5)\,sen(5x)\,dx\qquad\quad(i)}$


Fazendo a seguinte mudança de variável:

$\mathsf{cos(5x)=u\quad\Rightarrow\quad -5\,sen(5x)\,dx=du}$


Mudando os limites de integração:

$\begin{array}{lcl} \textsf{Quando }\mathsf{x=0}&\Rightarrow&\mathsf{u=cos(5\cdot 0)}\\\\ &\Rightarrow&\mathsf{u=cos(0)}\\\\ &\Rightarrow&\mathsf{u=1}\\\\\\ \textsf{Quando }\mathsf{x=4}&\Rightarrow&\mathsf{u=cos(5\cdot 4)}\\\\ &\Rightarrow&\mathsf{u=cos(20)} \end{array}$


Subsituindo em $\mathsf{(i)},$ a integral fica

$=\mathsf{\displaystyle-\,\frac{1}{5}\int_1^{cos(20)} u^2\,du}\\\\\\ =\mathsf{\displaystyle-\,\frac{1}{5}\cdot \frac{u^3}{3}\bigg|_1^{cos(20)}}\\\\\\ =\mathsf{\displaystyle-\,\frac{1}{15}\cdot \big[cos^3(20)-1\big]}\\\\\\\\ \therefore~~\boxed{\begin{array}{c}\mathsf{\displaystyle\int_0^4 sen(5x)\,cos^2(5x)\,dx=\frac{1-cos^3(20)}{15}} \end{array}}\qquad\quad\checkmark$


Bons estudos! :-)


Tags:  integral definida substituição mudança de variável função trigonométrica seno cosseno sen cos cálculo integral