What is the integral from 0 to x^3 of ln(t^2+1) ?

The integral from 0 to x^3 of ln(t^2+1) is x^3ln(x^6+1)-2(x^3-arctan(x^3))

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integral from 0 to x^3 of ln(t^2+1)

\int_{0}^{x^{3}} ln (t^{2} + 1) d t

x^{3} ln (x^{6} + 1) - 2 (x^{3} - arctan (x^{3}))

Solution steps

\int_{0}^{x^{3}} ln (t^{2} + 1) d t

Apply Integration By Parts

= [t ln (t^{2} + 1) - \int \frac{2 t ^{2}}{t ^{2} + 1} d t]_{0}^{x^{3}}

\int \frac{2 t ^{2}}{t ^{2} + 1} d t = 2 (- arctan (t) + t)

= [t ln (t^{2} + 1) - 2 (- arctan (t) + t)]_{0}^{x^{3}}

Compute the boundaries:

x^{3} ln (x^{6} + 1) - 2 (x^{3} - arctan (x^{3}))

= x^{3} ln (x^{6} + 1) - 2 (x^{3} - arctan (x^{3}))

What is the integral from 0 to x^3 of ln(t^2+1) ?
The integral from 0 to x^3 of ln(t^2+1) is x^3ln(x^6+1)-2(x^3-arctan(x^3))