Solve = ∫ (from 0 to n pi) of (sinx-cosx) wrt x

$$= \int _ { 0 } ^ { n \pi } ( \sin x - \cos x ) d x$$

Evaluate

$$-\sqrt{2}\sin(\frac{\pi \left(4n+1\right)}{4})+1$$

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.

$$-\cos(n\pi )-\sin(n\pi )-\left(-\cos(0)-\sin(0)\right)$$

Simplify.

$$-\sqrt{2}\sin(\frac{1}{4}\pi \left(4n+1\right))+1$$

Differentiate w.r.t. n

$$-\pi \sqrt{2}\cos(\frac{\pi \left(4n+1\right)}{4})$$