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

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

I-evaluate

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

Mga Hakbang ng Solution

I-multiply ang x at x para makuha ang x^{2}.

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

Suriin muna ang indefinite na integral.

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

I-factour out ang constant gamit ang \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

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

Simula \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} para sa k\neq -1, palitan ang \int x^{2}\mathrm{d}x ng \frac{x^{3}}{3}.

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

Pasimplehin.

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

Ang definite integral ay ang antiderivative ng expression na na-evaluate sa upper limit ng integration minus ang antiderivative na na-evaluate sa lower limit ng integration.

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

Pasimplehin.

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

I-differentiate ang w.r.t. n

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