Solve ∫ (from 0 to 05) of (528x+384x^2) wrt x=

$$\int _ { 0 } ^ { 05 } ( 528 x + 384 x ^ { 2 } ) d x =$$

Evaluate

$$0$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 528x+384x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 528x\mathrm{d}x+\int 384x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$528\int x\mathrm{d}x+384\int x^{2}\mathrm{d}x$$

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 528 times \frac{x^{2}}{2}.

$$264x^{2}+384\int x^{2}\mathrm{d}x$$

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 384 times \frac{x^{3}}{3}.

$$264x^{2}+128x^{3}$$

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.

$$264\times \left(0\times 5\right)^{2}+128\times \left(0\times 5\right)^{3}-\left(264\times 0^{2}+128\times 0^{3}\right)$$

Simplify.

$$0$$