Solve ∫ (from 5 to 100) of (x^2+1/x+x^4)=

$$\int _ { 5 } ^ { 100 } ( x ^ { 2 } + \frac { 1 } { x } + x ^ { 4 } ) =$$

Evaluate

$$\ln(20)+\frac{6000998000}{3}\approx 2000332669.662399054$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}+\frac{1}{x}+x^{4}\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int \frac{1}{x}\mathrm{d}x+\int x^{4}\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+\int \frac{1}{x}\mathrm{d}x+\int x^{4}\mathrm{d}x$$

Use \int \frac{1}{x}\mathrm{d}x=\ln(|x|) from the table of common integrals to obtain the result.

$$\frac{x^{3}}{3}+\ln(|x|)+\int x^{4}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}.

$$\frac{x^{3}}{3}+\ln(|x|)+\frac{x^{5}}{5}$$

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.

$$\frac{100^{3}}{3}+\ln(|100|)+\frac{100^{5}}{5}-\left(\frac{5^{3}}{3}+\ln(|5|)+\frac{5^{5}}{5}\right)$$

Simplify.

$$\frac{6000998000}{3}+\ln(20)$$