Question

Evaluate the integral.

∫₀π / 4tan ⁵ x ⁴ x d x
A) -7/24
B) 7/24
C) -1/24
D) 1/24

Answer 1

To evaluate the integral ∫₀π / 4tan⁵x⁴ dx, use the substitution method with u = tan(x). Simplify the expression and integrate each term separately, then substitute back u = tan(x) to find the answer.

Step-by-step explanation:

To evaluate the integral ∫₀π / 4tan⁵x⁴ dx, we can use the substitution method. Let u = tan(x). Then, du = sec²(x) dx. Rearranging the integral, we have ∫₀π / 4tan⁵x⁴ dx = ∫₀0 u⁵(1 + u²) du.

Using linear substitution, we can simplify this expression to ∫₀0 (1 + u²)u⁵ du = ∫₀0 (u⁵ + u⁷) du.

Now, integrating each term separately, we get (1/6)u⁶ + (1/8)u⁸. Substituting back u = tan(x), the final result of the integral becomes (1/6)tan⁶x + (1/8)tan⁸x, evaluated from 0 to π / 4. Plugging in these values and simplifying, we get the answer D) 1/24.