Solve ∫ (from 0 to frac { pi) of {2}}(1-sinθ)dθ

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } ( 1 - \sin \theta ) d \theta$$

Evaluate

$$\frac{\pi }{2}-1\approx 0.570796327$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1-\sin(\theta )\mathrm{d}\theta$$

Integrate the sum term by term.

$$\int 1\mathrm{d}\theta +\int -\sin(\theta )\mathrm{d}\theta$$

Factor out the constant in each of the terms.

$$\int 1\mathrm{d}\theta -\int \sin(\theta )\mathrm{d}\theta$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}\theta =a\theta .

$$\theta -\int \sin(\theta )\mathrm{d}\theta$$

Use \int \sin(\theta )\mathrm{d}\theta =-\cos(\theta ) from the table of common integrals to obtain the result. Multiply -1 times -\cos(\theta ).

$$\theta +\cos(\theta )$$

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.

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

Simplify.

$$\frac{\pi }{2}-1$$