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

The integral from 2 to sqrt(x of)tln(t) is -2ln(2)+(xln(x)-x)/4+1

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integral from 2 to sqrt(x of)tln(t)

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

- 2 ln (2) + \frac{x ln ( x ) - x}{4} + 1

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{t}{2} d t = \frac{t ^{2}}{4}

= [\frac{1}{2} t^{2} ln (t) - \frac{t ^{2}}{4}]_{2}^{x}

Compute the boundaries:

\frac{1}{4} x ln (x) - \frac{x}{4} - 2 ln (2) + 1

= \frac{1}{4} x ln (x) - \frac{x}{4} - 2 ln (2) + 1

Simplify

= - 2 ln (2) + \frac{x ln ( x ) - x}{4} + 1

\int_{0}^{1} x e^{\frac{x}{a}} d x

\int_{0}^{1} (x e^{x^{2}}) d x

\int_{- 1}^{3} (- x^{2}) d x

\int_{1}^{4} \frac{5}{x ^{3}} d x

\int_{0}^{1} y^{2} 3 (1 - y)^{2} d y

What is the integral from 2 to sqrt(x of)tln(t) ?
The integral from 2 to sqrt(x of)tln(t) is -2ln(2)+(xln(x)-x)/4+1