Integral F (x^3 + X)^5 (3x^2 + 1)dx Will Be _____? - Testpoint

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Integral f (x^3 + x)^5 (3x^2 + 1)dx will be _____?

(x^3 + x)^6/6 + C
(x^3 + x)^5/5 + C
(x^3 + x)^4/4 + C
None of these

Explanation

Substitution Method:
Let $u = x^{3} + x$ . Then, the derivative of $u$ is:
$\frac{d u}{d x} = 3 x^{2} + 1 ⟹ d u = (3 x^{2} + 1) d x$
Rewrite the Integral:
The integral becomes:
$\int u^{5} d u$
Integrate:
$\int u^{5} d u = \frac{u^{6}}{6} + C$
Substitute Back:
Replace $u$ with $x^{3} + x$ :
$\frac{(x^{3} + x)^{6}}{6} + C$

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