Solve ∫ (from 0 to frac { 3 pi) of {4}}(1-cosx)

$$\int _ { 0 } ^ { \frac { 3 \pi } { 4 } } ( 1 - \cos x )$$

Evaluate

$$\frac{3\pi }{4}-\frac{\sqrt{2}}{2}\approx 1.649087709$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1-\cos(x)\mathrm{d}x$$

Integrate the sum term by term.

$$\int 1\mathrm{d}x+\int -\cos(x)\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 1\mathrm{d}x-\int \cos(x)\mathrm{d}x$$

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

$$x-\int \cos(x)\mathrm{d}x$$

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

$$x-\sin(x)$$

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.

$$\frac{1}{4}\times 3\pi -\sin(\frac{1}{4}\times 3\pi )+\sin(0)$$

Simplify.

$$\frac{3\pi }{4}-\frac{\sqrt{2}}{2}$$