Solve ∫ (from 0 to 2 log x) of r-r^2dr

$$\int _ { 0 } ^ { 2 \log x } r - r ^ { 2 } d r$$

Evaluate

$$\frac{2\left(-4\log(x)+3\right)\log(x)^{2}}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int r-r^{2}\mathrm{d}r$$

Integrate the sum term by term.

$$\int r\mathrm{d}r+\int -r^{2}\mathrm{d}r$$

Factor out the constant in each of the terms.

$$\int r\mathrm{d}r-\int r^{2}\mathrm{d}r$$

Since \int r^{k}\mathrm{d}r=\frac{r^{k+1}}{k+1} for k\neq -1, replace \int r\mathrm{d}r with \frac{r^{2}}{2}.

$$\frac{r^{2}}{2}-\int r^{2}\mathrm{d}r$$

Since \int r^{k}\mathrm{d}r=\frac{r^{k+1}}{k+1} for k\neq -1, replace \int r^{2}\mathrm{d}r with \frac{r^{3}}{3}. Multiply -1 times \frac{r^{3}}{3}.

$$\frac{r^{2}}{2}-\frac{r^{3}}{3}$$

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.

$$\frac{1}{2}\times \left(2\ln(x)\ln(10)^{-1}\right)^{2}-\frac{1}{3}\times \left(2\ln(x)\ln(10)^{-1}\right)^{3}-\left(\frac{0^{2}}{2}-\frac{0^{3}}{3}\right)$$

Simplify.

$$\frac{2\log(x)^{2}\left(3-4\log(x)\right)}{3}$$

Differentiate w.r.t. x

$$\frac{4\log(e)\log(x)\left(-2\log(x)+1\right)}{x}$$