Solve ∫ (from 0 to 1) of x*3x^2 wrt x

$$\int _ { 0 } ^ { 1 } x \times 3 x ^ { 2 } d x$$

Evaluate

$$\frac{3}{4}=0.75$$

Solution Steps

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

$$\int _{0}^{1}x^{3}\times 3\mathrm{d}x$$

Evaluate the indefinite integral first.

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

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

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

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}.

$$\frac{3x^{4}}{4}$$

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.

$$\frac{3}{4}\times 1^{4}-\frac{3}{4}\times 0^{4}$$

Simplify.

$$\frac{3}{4}$$