Solve ∫ (from 1 to 2) of (3x^2+4x^3) wrt x

$$\int _ { 1 } ^ { 2 } ( 3 x ^ { 2 } + 4 x ^ { 3 } ) d x$$

Evaluate

$$22$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3x^{2}+4x^{3}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{2}\mathrm{d}x+\int 4x^{3}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{2}\mathrm{d}x+4\int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 3 times \frac{x^{3}}{3}.

$$x^{3}+4\int x^{3}\mathrm{d}x$$

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}. Multiply 4 times \frac{x^{4}}{4}.

$$x^{3}+x^{4}$$

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.

$$2^{3}+2^{4}-\left(1^{3}+1^{4}\right)$$

Simplify.

$$22$$