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Using the reduction formula for powers of tangent, evaluate I =…
Question
Using the reduction formula for powers of tangent, evaluate I = \int_0^{\pi/4} \tan^5 x\,dx and give its value rounded to 4 decimal places.
✓ Verified answer: 0.0966checked by our engine — not a guess
Step-by-step solution
Use T_n=\int_0^{\pi/4}\tan^n x\,dx with T_n = \frac{1}{n-1} - T_{n-2}, derived from \tan^n = \tan^{n-2}(\sec^2-1).
We need T_5 = \frac{1}{4} - T_3, and T_3 = \frac{1}{2} - T_1, with T_1=\int_0^{\pi/4}\tan x\,dx = [-\ln\cos x]_0^{\pi/4} = \frac{1}{2}\ln 2.
So T_3 = 1/2 - (1/2)\ln2, and T_5 = 1/4 - (1/2 - (1/2)\ln2) = -1/4 + (1/2)\ln2.
Numerically (1/2)\ln2 = 0.346574, minus 0.25 gives 0.096574.
Final answer0.0966
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