First slide

Methods of integration

Question

If $\int \tan^{7} x d x = f (x) + C$ then

Moderate

A

$f (x)$ is a polynomial of degree $8$ in $\tan x$
B

$f (x)$ is a polynomial of degree 5 in $\tan x$
C

$f (x)$ $= \frac{1}{6} \tan^{6} x - \frac{1}{4} \tan^{4} x + \frac{1}{2} \tan^{2} x + \log | \cos x | + C$
D

$f (x)$ is a polynomial of degree $6$ in $\tan x$

Solution

Putting $\tan x = t$ we have $d x = \frac{d t}{1 + t^{2}},$ So
$\int \tan^{7} x d x = \int \frac{t^{7}}{1 + t^{2}} d t$
$= \int (t^{5} - t^{3} + t - \frac{t}{1 + t^{2}}) d t$
$= \frac{t^{6}}{6} - \frac{t^{4}}{4} + \frac{t^{2}}{2} - \frac{1}{2} \log (1 + t^{2}) + C$
$= \frac{1}{6} \tan^{6} x - \frac{1}{4} \tan^{4} x + \frac{1}{2} \tan^{2} x$
$- \frac{1}{2} {logsec}^{2} x + C$
$\begin{aligned} = \frac{1}{6} \tan^{6} x - \frac{1}{4} \tan^{4} x + \frac{1}{2} \tan^{2} x \\ + \log | \cos x | + C . \end{aligned}$

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