First slide

Methods of integration

Question

$\int \frac{(9 \tan^{5} x + 7 \tan^{3} x) (1 + \tan^{2} x) d x}{{(\tan^{3} x + \tan x)}^{\frac{1}{2}}}$ =…

Difficult

A

$2 {(\tan x)}^{\frac{5}{2}} + c$
B

$2 {(\tan x)}^{\frac{7}{2}} + c$
C

$2 {(\tan x)}^{\frac{7}{2}} | \sec x | + c$
D

None

Solution

$t = \tan x, d t = \sec^{2} x d t$ $\int \frac{(9 t^{5} + 7 t^{3}) d t}{{(t^{3} + t)}^{1 / 2}}$ $\int \frac{9 t^{8} + 7 t^{6}}{{(t^{9} + t^{7})}^{1 / 2}} d t$ $y = t^{9} + t^{7}$
$\Rightarrow \int \frac{d y}{\sqrt{y}} \Rightarrow 2 \sqrt{y} + c$

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