Solve ∫ (from 10 to 15) of lnx^frac{x{2 wrt x

$$\int _ { 10 } ^ { 15 } \ln x ^ { \frac { x } { 2 } } d x$$

Evaluate

$$\frac{125\ln(5)}{4}+\frac{225\ln(3)}{4}-25\ln(2)-\frac{125}{8}\approx 79.138196487$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \log_{e}\left(x^{\frac{x}{2}}\right)\mathrm{d}x$$

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

$$\frac{\int \ln(x^{\frac{x}{2}})\mathrm{d}x}{\ln(e)}$$

Simplify.

$$\frac{\frac{\ln(x)x^{2}}{4}-\frac{x^{2}}{8}}{\ln(e)}$$

Simplify.

$$\frac{\ln(x)x^{2}}{4}-\frac{x^{2}}{8}$$

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.

$$\frac{1}{4}\ln(15)\times 15^{2}-\frac{15^{2}}{8}-\left(\frac{1}{4}\ln(10)\times 10^{2}-\frac{10^{2}}{8}\right)$$

Simplify.

$$-\frac{125}{8}+\frac{225\ln(3)}{4}+\frac{125\ln(5)}{4}-25\ln(2)$$