91Ó°ÊÓ

We want to evaluate the definite integral \( \int_{0}^{\pi / 4} \tan^{4} t \, dt \). Start by recognizing the need to simplify the integrand \( \tan^{4} t \). We can write \( \tan^{4} t \) as \( (\tan^2 t)^2 \) and use a trigonometric identity for simplification.

Recall the identity \( \tan^2 t = \sec^2 t - 1 \). Substitute this into the expression for \( \tan^4 t \): \[ \tan^4 t = (\sec^2 t - 1)^2 = \sec^4 t - 2\sec^2 t + 1. \] This will simplify the integrand for easier integration.

Split the integral into three separate integrals: \[ \int_{0}^{\pi / 4} \tan^4 t \, dt = \int_{0}^{\pi / 4} \sec^4 t \, dt - 2 \int_{0}^{\pi / 4} \sec^2 t \, dt + \int_{0}^{\pi / 4} 1 \, dt. \] Each of these integrals is simpler to solve individually.

To evaluate \( \int \sec^4 t \, dt \), use substitution or known results. For a general approach, use the reduction formula or integration by parts. We focus on the result obtained for definite integrals later if integration by parts is used explicitly here to find \( \int \sec^4 t \, dt = \frac{ \tan t \sec^2 t }{3} + \frac{2}{3} \tan t \). Evaluate this from 0 to \( \pi/4 \).

The integral \( \int \sec^2 t \, dt \) is a standard integral given by \( \tan t \). Therefore, \[ \int_{0}^{\pi / 4} \sec^2 t \, dt = [\tan t]_{0}^{\pi / 4} = \tan(\pi/4) - \tan(0) = 1 - 0 = 1. \] Apply this to the integral \(-2 \int_{0}^{\pi / 4} \sec^2 t \, dt = -2 \times 1 = -2.\)

This is a straightforward integral: \( \int 1 \, dt = t \). Evaluate from 0 to \( \pi/4 \): \[ \int_{0}^{\pi / 4} 1 \, dt = [t]_{0}^{\pi / 4} = \frac{\pi}{4} - 0 = \frac{\pi}{4} \].

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