Evaluate the following integrals: ^3 (x^4+1 )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x}^3\sin\big(\text{x}^4+1\big)\text{dx}$

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Answer

$\int\text{x}^3.\sin\big(\text{x}^4+1\big)\text{dx}$
Let $\text{x}^4+1=\text{t}$
$\Rightarrow4\text{x}^3=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{x}^3\text{ dx}=\frac{\text{dt}}{4}$
Now, $\int\text{x}^3.\sin\big(\text{x}^4+1\big)\text{dx}$
$=\frac{1}{4}\int\sin(\text{t})\text{dt}$
$=\frac{1}{4}[-\cos\text{t}]+\text{C}$
$=\frac{1}{4}[-\cos(\text{x}^4+1)]+\text{C}$

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