Evaluate the following integrals: ^3 ^4dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x}^3\cos\text{x}^4\text{dx}$

✓

Answer

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

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