Solve ∫ _{5pi/4}^2pi(cosx-sinx)

$$\int _ { \frac { 5 \pi } { 4 } } ^ { 2 \pi } ( \cos x - \sin x )$$

Evaluate

$$\sqrt{2}+1\approx 2.414213562$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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

$$\sin(x)+\cos(x)$$

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.

$$\sin(2\pi )+\cos(2\pi )-\left(\sin(\frac{1}{4}\times 5\pi )+\cos(\frac{1}{4}\times 5\pi )\right)$$

Simplify.

$$1+\sqrt{2}$$