Find the indefinite integral. $$\int \tan ^ { 5 } \left( \frac { x } { 5 } \right) d x$$
A) $\tan ^ { 2 } \left( \frac { x } { 5 } \right) \cdot \left( 1 + \frac { 5 } { 4 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) \right) + 5 \ln \left| \cos \left( \frac { x } { 5 } \right) \right| + C$
B)$\tan ^ { 2 } \left( \frac { x } { 5 } \right) \cdot \left( 1 - \frac { 5 } { 4 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) \right) - 5 \ln \left| \cos \left( \frac { x } { 5 } \right) \right| + C$
C) $\frac { 5 } { 2 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) \cdot \left( \frac { 1 } { 2 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) - 1 \right) - 5 \ln \left| \cos \left( \frac { x } { 5 } \right) \right| + C$
D) $\tan ^ { 2 } \left( \frac { x } { 5 } \right) \cdot \left( 1 + \frac { 5 } { 4 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) | - 5 \ln | \cos \left( \frac { x } { 5 } \right) \mid + C \right.$
E)$\frac { 5 } { 2 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) \cdot \left( 1 - \frac { 5 } { 4 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) - 1 \right) - 5 \ln \left| \cos \left( \frac { x } { 5 } \right) \right| + C$

Question

Find the indefinite integral. $$\int \tan ^ { 5 } \left( \frac { x } { 5 } \right) d x$$
A) $\tan ^ { 2 } \left( \frac { x } { 5 } \right) \cdot \left( 1 + \frac { 5 } { 4 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) \right) + 5 \ln \left| \cos \left( \frac { x } { 5 } \right) \right| + C$
B)$\tan ^ { 2 } \left( \frac { x } { 5 } \right) \cdot \left( 1 - \frac { 5 } { 4 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) \right) - 5 \ln \left| \cos \left( \frac { x } { 5 } \right) \right| + C$ 
C) $\frac { 5 } { 2 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) \cdot \left( \frac { 1 } { 2 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) - 1 \right) - 5 \ln \left| \cos \left( \frac { x } { 5 } \right) \right| + C$
D) $\tan ^ { 2 } \left( \frac { x } { 5 } \right) \cdot \left( 1 + \frac { 5 } { 4 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) | - 5 \ln | \cos \left( \frac { x } { 5 } \right) \mid + C \right.$
E)$\frac { 5 } { 2 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) \cdot \left( 1 - \frac { 5 } { 4 } \tan ^ { 2 } \left( \frac { x } { 5 } \right) - 1 \right) - 5 \ln \left| \cos \left( \frac { x } { 5 } \right) \right| + C$

Quizplus · Accepted Answer

The correct answer is obtained by recognizing that the integral of $\tan^5(x/5)$ involves breaking down the expression into lower powers of tangent and using trigonometric identities, such as $\tan^2(x) + 1 = \sec^2(x)$, and integrating by parts or substitution where necessary. Option C correctly represents the integral with the appropriate coefficients and terms, including the logarithmic term that arises from integrating $\tan(x)$ or $\sec^2(x)$ functions.

Find the Indefinite Integral ∫tan⁡5(x5)dx\int \tan ^ { 5 } \left( \frac { x } { 5 } \right) d x∫tan5(5x​)dx

Find the Indefinite Integral $\int \tan ^ { 5 } \left( \frac { x } { 5 } \right) d x$