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

Substitution Method:Let u=x3+xu=x3+x. Then, the derivative of uu is:dudx=3x2+1 ⟹ du=(3x2+1)dxdxdu=3x2+1⟹du=(3x2+1)dx

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 $x^{3} + x$ :
$\frac{(x^{3} + x)^{6}}{6} + C$