What is the integral from 0 to pi/4 of 4tan^4(x) ?

The integral from 0 to pi/4 of 4tan^4(x) is pi-8/3

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integral from 0 to pi/4 of 4tan^4(x)

\int_{0}^{\frac{π}{4}} 4 tan^{4} (x) d x

π - \frac{8}{3}

Decimal Notation

0.47492 \dots

Solution steps

\int_{0}^{\frac{π}{4}} 4 tan^{4} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int_{0}^{\frac{π}{4}} tan^{4} (x) d x

Apply Integral Reduction

= 4 ([\frac{tan ^{3} ( x )}{3}]_{0}^{\frac{π}{4}} - \int_{0}^{\frac{π}{4}} tan^{2} (x) d x)

\int_{0}^{\frac{π}{4}} tan^{2} (x) d x = - \frac{π}{4} + 1

= 4 ([\frac{tan ^{3} ( x )}{3}]_{0}^{\frac{π}{4}} - (- \frac{π}{4} + 1))

Simplify

4 ([\frac{tan ^{3} ( x )}{3}]_{0}^{\frac{π}{4}} - (- \frac{π}{4} + 1)) : 4 [\frac{1}{3} tan^{3} (x)]_{0}^{\frac{π}{4}} + π - 4

= 4 [\frac{1}{3} tan^{3} (x)]_{0}^{\frac{π}{4}} + π - 4

Compute the boundaries:

\frac{1}{3}

= 4 \cdot \frac{1}{3} + π - 4

Simplify

= π - \frac{8}{3}

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What is the integral from 0 to pi/4 of 4tan^4(x) ?
The integral from 0 to pi/4 of 4tan^4(x) is pi-8/3