Evaluate the following integrals: ^2e^ x^3 (e^ x^3 )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x}^2\text{e}^{\text{x}^3}\cos\big(\text{e}^{\text{x}^3}\big)\text{dx}$

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Answer

Let $\text{I}=\int\text{x}^2\text{e}^{\text{x}^3}\cos\big(\text{e}^{\text{x}^3}\big)\text{dx}\ ....(1)$

Let $\text{e}^{\text{x}^3}=\text{t}$ then,

$\text{d}\big(\text{e}^{\text{x}^3}\big)=\text{dt}$

$\Rightarrow3\text{x}^2\text{e}^{\text{x}^3}\text{dx}=\text{dt}$

$\Rightarrow\text{x}^2\text{e}^{\text{x}^3}\text{dx}=\frac{\text{dt}}{3}$

Putting $\text{e}^{\text{x}^3}=\text{t}$ and $\text{dx}=\frac{\text{dt}}{3}$ in equation (1), we get

$\text{I}=\int\cos\text{t}\frac{\text{dt}}{3}$

$=\frac{\sin\text{t}}{3}+\text{C}$

$=\frac{\sin\big(\text{e}^{\text{x}^3}\big)}{3}+\text{C}$

$\text{I}=\frac{1}{3}\sin\big(\text{e}^{\text{x}^3}\big)+\text{C}$

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