Evaluate the integral. The integral may not require integration by parts.
-$$\int x ^ { 4 } \sec x ^ { 5 } d x$$
A) $5 \ln \left| \sec x ^ { 5 } - \tan x ^ { 5 } \right| + C$
B) $x ^ { 6 } \sec x ^ { 6 } + C$
C) $\frac { 1 } { 4 } \ln \left| \sec x ^ { 5 } + \tan x ^ { 5 } \right| + C$
D) $\frac { 1 } { 5 } \ln \left| \sec x ^ { 5 } + \tan x ^ { 5 } \right| + C$

Question

Quizplus · Accepted Answer

This integral does not directly match any standard integral form and requires a substitution to simplify. Let $u = x^5$, then $du = 5x^4 dx$. Adjusting for the $5$, the integral becomes $\frac{1}{5}\int \sec(u) du$, which integrates to $\frac{1}{5} \ln | \sec(u) + \tan(u) | + C$. Substituting back $u = x^5$, the answer is $\frac{1}{5} \ln | \sec(x^5) + \tan(x^5) | + C$.

Evaluate the Integral ∫x4sec⁡x5dx\int x ^ { 4 } \sec x ^ { 5 } d x∫x4secx5dx

Evaluate the Integral $\int x ^ { 4 } \sec x ^ { 5 } d x$