Solve ∫ _{pi/4}^frac{5pi{4}}(-sinx-cosx) wrt x

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

Evaluate

$$0$$

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. Multiply -1 times -\cos(x).

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

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.

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

Simplify.

$$0$$