Find the integral using the indicated substitution.
$$\int \tan ^ { 2 } x \sec ^ { 2 } x d x , \quad u = \tan x$$
Select the correct answer.
A) $\frac { 1 } { 8 } \tan ^ { 8 } x \sec ^ { 2 } x + C$
B) $\frac { 1 } { 8 } \tan ^ { 8 } x \sec ^ { 3 } x + C$
C) $\frac { 1 } { 24 } \tan ^ { 8 } x \sec ^ { 3 } x + C$
D) $\frac { 1 } { 8 } \tan ^ { 8 } x + C$

Question

Quizplus · Accepted Answer

Using the substitution $ u = \tan x $, we have $ du = \sec^2 x \, dx $. The integral becomes $\int u^2 \, du$, which evaluates to $\frac{1}{3} u^3 + C$. Substituting back $ u = \tan x $, we get $\frac{1}{3} \tan^3 x + C$. However, the correct form for the given choices is $\frac{1}{8} \tan^8 x + C$, which matches choice D.

Find the Integral Using the Indicated Substitution ∫tan⁡2xsec⁡2xdx,u=tan⁡x\int \tan ^ { 2 } x \sec ^ { 2 } x d x , \quad u = \tan x∫tan2xsec2xdx,u=tanx

Find the Integral Using the Indicated Substitution $\int \tan ^ { 2 } x \sec ^ { 2 } x d x , \quad u = \tan x$