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

The integral from 2 to x^3 of ln(t^2) is 6x^3ln(x)-2x^3-4ln(2)+4

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

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

6 x^{3} ln (x) - 2 x^{3} - 4 ln (2) + 4

Solution steps

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

Apply Integration By Parts

: [t ln (t^{2}) - \int 2 d t]_{2}^{x^{3}}

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

\int 2 d t = 2 t

= [t ln (t^{2}) - 2 t]_{2}^{x^{3}}

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= [2 t ln (t) - 2 t]_{2}^{x^{3}}

Compute the boundaries:

6 x^{3} ln (x) - 2 x^{3} - 4 ln (2) + 4

= 6 x^{3} ln (x) - 2 x^{3} - 4 ln (2) + 4

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\int_{5}^{2 x} ln (u) d u

What is the integral from 2 to x^3 of ln(t^2) ?
The integral from 2 to x^3 of ln(t^2) is 6x^3ln(x)-2x^3-4ln(2)+4