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

The integral from 0 to pi/4 of sec^3(θ) is (sqrt(2)+ln(1+sqrt(2)))/2

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integral from 0 to pi/4 of sec^3(θ)

\int_{0}^{\frac{π}{4}} sec^{3} (θ) d θ

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

Decimal Notation

1.14779 \dots

Solution steps

\int_{0}^{\frac{π}{4}} sec^{3} (θ) d θ

Apply Integral Reduction

= [\frac{sec ^{2} ( θ ) sin ( θ )}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} \cdot \int_{0}^{\frac{π}{4}} sec (θ) d θ

\int_{0}^{\frac{π}{4}} sec (θ) d θ = ln (1 + 2)

= [\frac{sec ^{2} ( θ ) sin ( θ )}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2)

Simplify

[\frac{sec ^{2} ( θ ) sin ( θ )}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2) : [\frac{1}{2} sec (θ) tan (θ)]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2)

= [\frac{1}{2} sec (θ) tan (θ)]_{0}^{\frac{π}{4}} + \frac{1}{2} ln (1 + 2)

Compute the boundaries:

\frac{1}{2}

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

Simplify

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

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What is the integral from 0 to pi/4 of sec^3(θ) ?
The integral from 0 to pi/4 of sec^3(θ) is (sqrt(2)+ln(1+sqrt(2)))/2