If ∫x71+x42dx=14log⁡1+x4+f(x)+C then

A
$f (x) = 1 + x^{4}$
B
$f (x) = \frac{1}{{(1 + x^{4})}^{2}}$
C
$f (x) = \frac{1}{1 + x^{4}}$
D
$f (x) = \tan^{- 1} (1 + x^{4})$

Solution:

Put $1 + x^{4} = t$ , so that

$\begin{array}{l} \int \frac{x^{7}}{{(1 + x^{4})}^{2}} d x = \frac{1}{4} \int \frac{(t - 1)}{t^{2}} d t \\ = \frac{1}{4} \log | t | + \frac{1}{4} \frac{1}{t} + C \\ = \frac{1}{4} [\log (1 + x^{4}) + \frac{1}{1 + x^{4}}] + C \\ ∴ f (x) = \frac{1}{1 + x^{4}} \end{array}$

If $\int \frac{x^{7}}{{(1 + x^{4})}^{2}} d x = \frac{1}{4} [\log (1 + x^{4}) + f (x)] + C$ then

Solution:

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