Question

intergrate the following x^4 sin ( x^5 + 6 ) dx is equal to​

Answer 1

Given ,

The function is

• $\tt f(x) = {x}^{4} sin( {x}^{5} + 6)$

Let ,

$\tt {x}^{5} + 6 = t$

Differentiaitng t wrt x , we get

$\tt \implies 5 {x}^{4} = \frac{dt}{dx}$

$\tt \implies dx = \frac{dt}{5 {x}^{4} }$

Thus ,

$\tt \implies \int{ {x}^{4} sin( {x}^{5} + 6)} \: \: dx$

$\tt \implies \int{ {x}^{4} sin(t)} \: \: \frac{dt}{5 {x}^{4} }$

$\tt \implies \int{ \frac{sin(t)}{5} } \: \: dt$

$\tt \implies \frac{1}{5} \int{ sin(t)} \: \: dt$

$\tt \implies - \frac{1}{5} \times cos(t) + C$

$\tt \implies - \frac{1}{5} \times cos( {x}^{5} + 6)+ C$

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