Solve ∫ (from 1 to 4) of (x^3+5/sqrt{x}) wrt x

$$\int _ { 1 } ^ { 4 } ( x ^ { 3 } + \frac { 5 } { \sqrt { x } } ) d x$$

Evaluate

$$\frac{295}{4}=73.75$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{3}+\frac{5}{\sqrt{x}}\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{3}\mathrm{d}x+\int \frac{5}{\sqrt{x}}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{3}\mathrm{d}x+5\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}.

$$\frac{x^{4}}{4}+5\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

Rewrite \frac{1}{\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{1}{2}}}{\frac{1}{2}}. Simplify and convert from exponential to radical form. Multiply 5 times 2\sqrt{x}.

$$\frac{x^{4}}{4}+10\sqrt{x}$$

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.

$$\frac{4^{4}}{4}+10\times 4^{\frac{1}{2}}-\left(\frac{1^{4}}{4}+10\times 1^{\frac{1}{2}}\right)$$

Simplify.

$$\frac{295}{4}$$