Question

Integration of 0 to pi tan^7x DX​

Answer 1

0.

Explanation:

try this suggested solution, the answer is marked with colour.

Answer 2

0

Explanation:

You want the integral of the 7th power of the tangent function over one full period.

Symmetry

The tangent function is an odd function, meaning it is symmetrical about the origin. The 7th power preserves the sign of the function, so tan⁷(x) is also an odd function symmetrical about the origin.

The tangent function has periodicity π, so it will have the same behavior relative to x=π that it has relative to x=0. In short, the function is a reflection of itself about the point (π/2, 0), the middle of the interval of integration.

The integral from 0 to π/2 will be canceled by the integral from π/2 to π. Hence the integral over 0 to π is zero.