Solve ∫ _{-n/2}^frac{1{2}}2x^2*y/2x wrt x

$$\int _ { - \frac { n } { 2 } } ^ { \frac { 1 } { 2 } } 2 x ^ { 2 } \cdot y / 2 x d x$$

Evaluate

$$\frac{y\left(1-n^{4}\right)}{64}$$

Solution Steps

Cancel out 2 and 2.

$$\int _{-\frac{n}{2}}^{\frac{1}{2}}x^{2}yx\mathrm{d}x$$

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

$$\int _{-\frac{n}{2}}^{\frac{1}{2}}x^{3}y\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int x^{3}y\mathrm{d}x$$

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

$$y\int x^{3}\mathrm{d}x$$

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}.

$$y\times \frac{x^{4}}{4}$$

Simplify.

$$\frac{yx^{4}}{4}$$

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.

$$\frac{1}{4}y\times \left(\frac{1}{2}\right)^{4}-\frac{1}{4}y\left(-\frac{1}{2}n\right)^{4}$$

Simplify.

$$\frac{y-yn^{4}}{64}$$

Differentiate w.r.t. y

$$\frac{1-n^{4}}{64}$$