Solve ∫ (from 0 to 1) of ln(1+p^2)d(1+p^2)

$$\int _ { 0 } ^ { 1 } \ln ( 1 + p ^ { 2 } ) d ( 1 + p ^ { 2 } )$$

Evaluate

$$d\left(p^{2}+1\right)\ln(p^{2}+1)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int d\left(1+p^{2}\right)\log_{e}\left(1+p^{2}\right)\mathrm{d}x$$

Find the integral of d\left(1+p^{2}\right)\log_{e}\left(1+p^{2}\right) using the table of common integrals rule \int a\mathrm{d}x=ax.

$$d\left(1+p^{2}\right)\log_{e}\left(1+p^{2}\right)x$$

Simplify.

$$\ln(1+p^{2})d\left(1+p^{2}\right)x$$

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.

$$\ln(1+p^{2})d\left(1+p^{2}\right)\times 1-\ln(1+p^{2})d\left(1+p^{2}\right)\times 0$$

Simplify.

$$\ln(1+p^{2})d\left(1+p^{2}\right)$$

Differentiate w.r.t. p

$$2dp\left(\ln(p^{2}+1)+1\right)$$