What is the integral from 0 to pi/4 of 1/(cos^4(x)) ?

The integral from 0 to pi/4 of 1/(cos^4(x)) is 4/3

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integral from 0 to pi/4 of 1/(cos^4(x))

\int_{0}^{\frac{π}{4}} \frac{1}{cos ^{4} ( x )} d x

\frac{4}{3}

Decimal Notation

1.33333 \dots

Solution steps

\int_{0}^{\frac{π}{4}} \frac{1}{cos ^{4} ( x )} d x

Rewrite using trig identities

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

Apply u-substitution

: \int_{0}^{1} 1 + u^{2} d u

= \int_{0}^{1} 1 + u^{2} 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

= \int_{0}^{1} 1 d u + \int_{0}^{1} u^{2} d u

\int_{0}^{1} 1 d u = 1

\int_{0}^{1} u^{2} d u = \frac{1}{3}

= 1 + \frac{1}{3}

Simplify

1 + \frac{1}{3} : \frac{4}{3}

= \frac{4}{3}

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What is the integral from 0 to pi/4 of 1/(cos^4(x)) ?
The integral from 0 to pi/4 of 1/(cos^4(x)) is 4/3