Question

integral x⁴sin(x⁵+6) dx is equal to
​

Answer 1

$\underline{\textsf{To find:}}$

$\mathsf{\displaystyle\int\,x^4\,sin(x^5+6)\,dx}$

$\underline{\textsf{Solution:}}$

$\textsf{We apply change of variable method to solve the given integral}$

$\textsf{Consider,}$

$\mathsf{\displaystyle\int\,x^4\,sin(x^5+6)\,dx}$

$\textsf{Take,}\;\mathsf{t=x^5+6}$

$\mathsf{\dfrac{dt}{dx}=5x^4}$

$\implies\mathsf{\dfrac{dt}{5}=x^4\,dx}$

$\textsf{Now,}$

$\mathsf{\displaystyle\int\,x^4\,sin(x^5+6)\,dx}$

$\mathsf{=\displaystyle\int\,sin(x^5+6)\,(x^4dx)}$

$\mathsf{=\displaystyle\int\,sint\,\dfrac{dt}{5}}$

$\mathsf{=\displaystyle\dfrac{1}{5}\int\,sint\,dt}$

$\mathsf{=\dfrac{1}{5}(-cost)+C}$

$\mathsf{=\dfrac{-cos(x^5+6)}{5}+C}$

$\implies\boxed{\mathsf{\displaystyle\int\,x^4\,sin(x^5+6)\,dx=\dfrac{-cos(x^5+6)}{5}+C}}$

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