Solve ∫ _{pi/4}^pileft(sin(x)-cos(x)right) wrt x

$$\int_{ \frac{ \pi }{ 4 } }^{ \pi } \left( \sin ( x ) - \cos ( x ) \right) d x$$

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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

$$-\cos(x)-\sin(x)$$

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.

$$-\cos(\pi )-\sin(\pi )-\left(-\cos(\frac{1}{4}\pi )-\sin(\frac{1}{4}\pi )\right)$$

Simplify.

$$1+\sqrt{2}$$