Question

$$int \tan^{9}xsec^{2}x dx$$
a.) $\frac{1}{10}\tan^{10}x + c$
b.) $\frac{1}{10}sec^{10}x + c$
c.) $\frac{1}{9}sec^{10}x + c$
d.) $\frac{1}{9}\tan^{10}x + c$

Explanation:

Step1: Choose substitution variable

Let $u = \tan x$

Step2: Find derivative of u

$\frac{du}{dx} = \sec^2 x \implies du = \sec^2 x dx$

Step3: Rewrite integral in terms of u

$\int u^9 du$

Step4: Apply power rule for integration

$\frac{u^{9+1}}{9+1} + C = \frac{u^{10}}{10} + C$

Step5: Substitute back $u=\tan x$

$\frac{1}{10}\tan^{10}x + C$

Answer:

a.) $\frac{1}{10}\tan^{10}x + c$