What is the integral from 1 to 4 of sqrt(x)ln(x^9) ?

The integral from 1 to 4 of sqrt(x)ln(x^9) is -28+96ln(2)

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integral from 1 to 4 of sqrt(x)ln(x^9)

\int_{1}^{4} x ln (x^{9}) d x

- 28 + 96 ln (2)

Solution steps

\int_{1}^{4} x ln (x^{9}) d x

Simplify

ln (x^{9}) : 9 ln (x)

= \int_{1}^{4} 9 x ln (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 9 \cdot \int_{1}^{4} x ln (x) d x

Apply Integration By Parts

: [\frac{2}{3} x^{\frac{3}{2}} ln (x) - \int \frac{2}{3} x^{\frac{1}{2}} d x]_{1}^{4}

= 9 [\frac{2}{3} x^{\frac{3}{2}} ln (x) - \int \frac{2}{3} x^{\frac{1}{2}} d x]_{1}^{4}

\int \frac{2}{3} x^{\frac{1}{2}} d x = \frac{4}{9} x^{\frac{3}{2}}

= 9 [\frac{2}{3} x^{\frac{3}{2}} ln (x) - \frac{4}{9} x^{\frac{3}{2}}]_{1}^{4}

Compute the boundaries:

- \frac{28}{9} + \frac{32}{3} ln (2)

= 9 (- \frac{28}{9} + \frac{32}{3} ln (2))

Simplify

= - 28 + 96 ln (2)

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What is the integral from 1 to 4 of sqrt(x)ln(x^9) ?
The integral from 1 to 4 of sqrt(x)ln(x^9) is -28+96ln(2)