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

The integral from 0 to pi/4 of 3tan^3(x) is 3(-1/2 ln(2)+1/2)

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integral from 0 to pi/4 of 3tan^3(x)

\int_{0}^{\frac{π}{4}} 3 tan^{3} (x) d x

3 (- \frac{1}{2} ln (2) + \frac{1}{2})

Decimal

0.46027 \dots

Solution steps

\int_{0}^{\frac{π}{4}} 3 tan^{3} (x) d x

Take the constant out:

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

= 3 \cdot \int_{0}^{\frac{π}{4}} tan^{3} (x) d x

Rewrite using trig identities

= 3 \cdot \int_{0}^{\frac{π}{4}} (- 1 + sec^{2} (x)) tan (x) d x

Apply u-substitution

= 3 \cdot \int_{1}^{2} \frac{- 1 + u ^{2}}{u} d u

Expand

\frac{- 1 + u ^{2}}{u} : - \frac{1}{u} + u

= 3 \cdot \int_{1}^{2} - \frac{1}{u} + u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (- \int_{1}^{2} \frac{1}{u} d u + \int_{1}^{2} u d u)

\int_{1}^{2} \frac{1}{u} d u = \frac{1}{2} ln (2)

\int_{1}^{2} u d u = \frac{1}{2}

= 3 (- \frac{1}{2} ln (2) + \frac{1}{2})

What is the integral from 0 to pi/4 of 3tan^3(x) ?
The integral from 0 to pi/4 of 3tan^3(x) is 3(-1/2 ln(2)+1/2)