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

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

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

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

\frac{4}{3}

Decimal Notation

1.33333 \dots

Solution steps

\int_{0}^{\frac{π}{4}} sec^{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}

ln 3 x^{3} \cdot sec^{3} ({5 x + 4})

\int \frac{x + 3}{x ^{2} + 2 x} d x

\frac{\partial}{\partial x} (2 x^{3} y)

y^{^{'}} = - 2 y

f (x) = (x^{2} - 1)^{2}, (- 2, 9)

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