Solve ∫ (from 4 to 9) of sqrt{x}(1+5) wrt x

$$\int _ { 4 } ^ { 9 } \sqrt { x } ( 1 + 5 ) d x$$

Evaluate

$$76$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{x}\left(1+5\right)\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\left(1+5\right)\int \sqrt{x}\mathrm{d}x$$

Rewrite \sqrt{x} as x^{\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{3}{2}}}{\frac{3}{2}}. Simplify.

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

Simplify.

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

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.

$$4\times 9^{\frac{3}{2}}-4\times 4^{\frac{3}{2}}$$

Simplify.

$$76$$