Solve ∫ (from 0 to pi / 4) of f(cosθ)dθ

$$\int _ { 0 } ^ { \pi / 4 } f ( \cos \theta ) d \theta$$

Evaluate

$$\frac{\sqrt{2}f}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int f\cos(\theta )\mathrm{d}\theta$$

Factor out the constant using \int af\left(\theta \right)\mathrm{d}\theta =a\int f\left(\theta \right)\mathrm{d}\theta .

$$f\int \cos(\theta )\mathrm{d}\theta$$

Use \int \cos(f)\mathrm{d}f=\sin(f) from the table of common integrals to obtain the result.

$$f\sin(\theta )$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$f\sin(\frac{1}{4}\pi )-f\sin(0)$$

Simplify.

$$\frac{f\sqrt{2}}{2}$$

Differentiate w.r.t. f

$$\frac{\sqrt{2}}{2} = 0.7071067811865476$$