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

The integral from sqrt(x) to x^2 of ln(t) is 2x^2ln(x)-x^2-(sqrt(x)ln(x))/2+sqrt(x)

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

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

2 x^{2} ln (x) - x^{2} - \frac{x ln ( x )}{2} + x

Solution steps

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

Apply Integration By Parts

: [t ln (t) - \int 1 d t]_{x}^{x^{2}}

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

\int 1 d t = t

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

Compute the boundaries:

2 x^{2} ln (x) - x^{2} - \frac{1}{2} x ln (x) + x

= 2 x^{2} ln (x) - x^{2} - \frac{1}{2} x ln (x) + x

Simplify

= 2 x^{2} ln (x) - x^{2} - \frac{x ln ( x )}{2} + x

\int_{0}^{5} 9 - x^{2} d x

\int_{\frac{1}{4}}^{2} ln (x) d x

\int_{0}^{1} \frac{4 x ^{2}}{1 + x ^{2}} d x

\int_{2}^{3} - 72 x^{- 3} d x

\int_{0}^{4} 1000 x + 250 d x

What is the integral from sqrt(x) to x^2 of ln(t) ?
The integral from sqrt(x) to x^2 of ln(t) is 2x^2ln(x)-x^2-(sqrt(x)ln(x))/2+sqrt(x)