Solve ∫ (from 0 to 4) of 3x^2+4x wrt x

$$\int _ { 0 } ^ { 4 } 3 x ^ { 2 } + 4 x d x$$

Evaluate

$$96$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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}. Multiply 4 times \frac{x^{2}}{2}.

$$x^{3}+2x^{2}$$

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.

$$4^{3}+2\times 4^{2}-\left(0^{3}+2\times 0^{2}\right)$$

Simplify.

$$96$$