If ∫dx1+x+13=32(x+1)2/3−3(x+1)1/3+f(x)+C then f(x) is equal to

If $\int \frac{d x}{1 + \sqrt[3]{x + 1}} = \frac{3}{2} (x + 1)^{2 / 3} - 3 (x + 1)^{1 / 3}$
$+ f (x) + C then f (x)$ is equal to

A
$\log | 1 + \sqrt[3]{x + 1} |$
B
$3 \log | 1 + \sqrt[3]{x + 1} |$
C
$\frac{2}{3} \log | 1 + \sqrt[3]{x + 1} |$
D
$\frac{1}{3} \log | 1 + \sqrt[3]{x + 1} |$

Solution:

Put $x + 1 = t^{3}$ , we have

$\int \frac{d x}{1 + \sqrt[3]{x + 1}} = \int \frac{3 t^{2} d t}{1 + t} = 3 \int (t - 1 + \frac{1}{1 + t}) d t$

$\begin{aligned} = 3 [\frac{t^{2}}{2} - t + \log | 1 + t |] + C \\ = \frac{3}{2} (x + 1)^{2 / 3} - 3 (x + 1)^{1 / 3} + \end{aligned}$

$3 \log | 1 + \sqrt[3]{x + 1} | + C$

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