Solve ∫ (from 1 to 5) of 3x+4 wrt x=

$$\int _ { 1 } ^ { 5 } 3 x + 4 d x =$$

Evaluate

$$52$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

Find the integral of 4 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{3x^{2}}{2}+4x$$

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.

$$\frac{3}{2}\times 5^{2}+4\times 5-\left(\frac{3}{2}\times 1^{2}+4\times 1\right)$$

Simplify.

$$52$$