Solve ∫ (from 0 to pi) of x*cosnsqrt{pi}x wrt x

$$\int _ { 0 } ^ { \pi } x \cdot \cos n \sqrt { \pi } x d x$$

Evaluate

$$\frac{\pi ^{\frac{7}{2}}\cos(n)}{3}$$

Solution Steps

Multiply x and x to get x^{2}.

$$\int _{0}^{\pi }x^{2}\cos(n)\sqrt{\pi }\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int x^{2}\cos(n)\sqrt{\pi }\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\cos(n)\sqrt{\pi }\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}.

$$\cos(n)\sqrt{\pi }\times \frac{x^{3}}{3}$$

Simplify.

$$\frac{\sqrt{\pi }\cos(n)x^{3}}{3}$$

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.

$$\frac{1}{3}\pi ^{\frac{1}{2}}\cos(n)\pi ^{3}-\frac{1}{3}\pi ^{\frac{1}{2}}\cos(n)\times 0^{3}$$

Simplify.

$$\frac{\cos(n)\pi ^{\frac{7}{2}}}{3}$$

Differentiate w.r.t. n

$$-\frac{\pi ^{\frac{7}{2}}\sin(n)}{3}$$