Solve ∫ (from 0 to frac { pi) of {2}}f(sinx) wrt x=

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } f ( \sin x ) d x =$$

Evaluate

$$f$$

Solution Steps

Evaluate the indefinite integral first.

$$\int f\sin(x)\mathrm{d}x$$

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

$$f\int \sin(x)\mathrm{d}x$$

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

$$-f\cos(x)$$

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.

$$-f\cos(\frac{1}{2}\pi )+f\cos(0)$$

Simplify.

$$f$$

Differentiate w.r.t. f

$$1$$