Solve ∫ (from 2 to 3) of xcos3y^4 wrt x

$$\int _ { 2 } ^ { 3 } x \cos 3 y ^ { 4 } d x$$

Evaluate

$$\frac{5\cos(3y^{4})}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x\cos(3y^{4})\mathrm{d}x$$

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

$$\cos(3y^{4})\int x\mathrm{d}x$$

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

$$\cos(3y^{4})\times \frac{x^{2}}{2}$$

Simplify.

$$\frac{\cos(3y^{4})x^{2}}{2}$$

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.

$$\frac{1}{2}\cos(3y^{4})\times 3^{2}-\frac{1}{2}\cos(3y^{4})\times 2^{2}$$

Simplify.

$$\frac{5\cos(3y^{4})}{2}$$

Differentiate w.r.t. y

$$-30y^{3}\sin(3y^{4})$$