Solve ∫ _{1/5}^7ln(5)x wrt x

$$\int_{ \frac{ 1 }{ 5 } }^{ 7 } \ln ( 5 ) x d x$$

Evaluate

$$\frac{612\ln(5)}{25}\approx 39.399040096$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x\log_{e}\left(5\right)\mathrm{d}x$$

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

$$\log_{e}\left(5\right)\int x\mathrm{d}x$$

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

$$\log_{e}\left(5\right)\times \frac{x^{2}}{2}$$

Simplify.

$$\frac{\ln(5)x^{2}}{2}$$

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.

$$\frac{1}{2}\ln(5)\times 7^{2}-\frac{1}{2}\ln(5)\times \left(\frac{1}{5}\right)^{2}$$

Simplify.

$$\frac{612\ln(5)}{25}$$