Solve ∫ (from 0 to frac { pi) of {4}}2x^2ny2x wrt x

$$\int _ { 0 } ^ { \frac { \pi } { 4 } } 2 x ^ { 2 } n y 2 x d x$$

Evaluate

$$\frac{\pi ^{4}ny_{2}}{512}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\int _{0}^{\frac{\pi }{4}}2x^{3}ny_{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 2x^{3}ny_{2}\mathrm{d}x$$

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

$$2ny_{2}\int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}.

$$2ny_{2}\times \frac{x^{4}}{4}$$

Simplify.

$$\frac{ny_{2}x^{4}}{2}$$

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.

$$\frac{1}{2}ny_{2}\times \left(\frac{1}{4}\pi \right)^{4}-\frac{1}{2}ny_{2}\times 0^{4}$$

Simplify.

$$\frac{ny_{2}\pi ^{4}}{512}$$

Differentiate w.r.t. n

$$\frac{\pi ^{4}y_{2}}{512}$$