Solve ∫ (from 0 to 4) of (2x+1)3/2 wrt x

$$\int _ { 0 } ^ { 4 } ( 2 x + 1 ) \frac { 3 } { 2 } d x$$

Evaluate

$$30$$

Solution Steps

Use the distributive property to multiply 2x+1 by \frac{3}{2}.

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

Cancel out 2 and 2.

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

Evaluate the indefinite integral first.

$$\int 3x+\frac{3}{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x\mathrm{d}x+\int \frac{3}{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x\mathrm{d}x+\int \frac{3}{2}\mathrm{d}x$$

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

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

Find the integral of \frac{3}{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

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

Simplify.

$$30$$