Question

Qual a integral de x³ cos x² dx?

Answer 1

$\large\begin{array}{l}\mathsf{I=\displaystyle\int\!x^3\,cos(x^2)\,dx}\\\\ =\mathsf{\displaystyle\int\!x^2\cdot x\,cos(x^2)\,dx}\\\\ =\mathsf{\displaystyle\int\!\frac{\,1\,}{2}\,x^2\cdot 2x\,cos(x^2)\,dx}\\\\ =\mathsf{\displaystyle\int\!\frac{\,1\,}{2}\,x^2\,cos(x^2)\cdot 2x\,dx\qquad\quad(i)} \end{array}$


$\large\begin{array}{l} \textsf{M\'etodo de integra\c{c}\~ao por partes:}\\\\ \begin{array}{lcl} \mathsf{u=\dfrac{\,1\,}{2}\,x^2}&~\Rightarrow~&\mathsf{du=x\,dx}\\\\ \mathsf{dv=cos(x^2)\cdot 2x\,dx}&~\Leftarrow~&\mathsf{v=sen(x^2)} \end{array} \end{array}$


$\large\begin{array}{l} \mathsf{\displaystyle\int\!u\,dv=u\cdot v-\int\!v\,du}\\\\ \mathsf{\displaystyle\int\!\frac{\,1\,}{2}\,x^2\,cos(x^2)\cdot 2x\,dx=\frac{\,1\,}{2}\,x^2\cdot sen(x^2)-\int\!sen(x^2)\cdot x\,dx}\\\\ \mathsf{\displaystyle\int\!x^3\,cos(x^2)\,dx=\frac{\,1\,}{2}\,x^2\,sen(x^2)-\int\!sen(x^2)\cdot \frac{\,1\,}{2}\cdot 2x\,dx}\\\\ \mathsf{\displaystyle\int\!x^3\,cos(x^2)\,dx=\frac{\,1\,}{2}\,x^2\,sen(x^2)-\frac{\,1\,}{2}\int\!sen(x^2)\cdot 2x\,dx} \end{array}$

$\large\begin{array}{l} \mathsf{\displaystyle\int\!x^3\,cos(x^2)\,dx=\frac{\,1\,}{2}\,x^2\,sen(x^2)-\frac{\,1\,}{2}\int\!sen\,w\,dw\qquad(w=x^2)}\\\\ \mathsf{\displaystyle\int\!x^3\,cos(x^2)\,dx=\frac{\,1\,}{2}\,x^2\,sen(x^2)-\frac{\,1\,}{2}\cdot (-cos\,w)+C}\\\\ \mathsf{\displaystyle\int\!x^3\,cos(x^2)\,dx=\frac{\,1\,}{2}\,x^2\,sen(x^2)+\frac{\,1\,}{2}\,cos\,w+C}\\\\\\ \therefore~~\boxed{\begin{array}{c} \mathsf{\displaystyle\int\!x^3\,cos(x^2)\,dx=\frac{\,1\,}{2}\,x^2\,sen(x^2)+\frac{\,1\,}{2}\,cos(x^2)+C} \end{array}}\qquad\checkmark \end{array}$


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$\large\begin{array}{l} \textsf{D\'uvidas? Comente.}\\\\\\ \textsf{Bons estudos! :-)} \end{array}$


Tags: integral indefinida integração por partes produto cosseno cos substituição cálculo integral