What is the integral from 1 to 5 of sqrt(2x)ln(x^3) ?

The integral from 1 to 5 of sqrt(2x)ln(x^3) is sqrt(2)(10sqrt(5)ln(5)+(-20sqrt(5)+4)/3)

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integral from 1 to 5 of sqrt(2x)ln(x^3)

\int_{1}^{5} 2 x ln (x^{3}) d x

2 (105 ln (5) + \frac{- 20 5 + 4}{3})

Decimal

31.69866 \dots

Solution steps

\int_{1}^{5} 2 x ln (x^{3}) d x

2 x = 2 x

= \int_{1}^{5} 2 x ln (x^{3}) d x

Take the constant out:

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

= 2 \cdot \int_{1}^{5} x ln (x^{3}) d x

Apply Integration By Parts

= 2 [\frac{2}{3} x^{\frac{3}{2}} ln (x^{3}) - \int 2 x^{\frac{1}{2}} d x]_{1}^{5}

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

= 2 [\frac{2}{3} x^{\frac{3}{2}} ln (x^{3}) - \frac{4}{3} x^{\frac{3}{2}}]_{1}^{5}

\frac{2}{3} x^{\frac{3}{2}} ln (x^{3}) = 2 x^{\frac{3}{2}} ln (x)

= 2 [2 x^{\frac{3}{2}} ln (x) - \frac{4}{3} x^{\frac{3}{2}}]_{1}^{5}

Compute the boundaries:

105 ln (5) - \frac{20 5}{3} + \frac{4}{3}

= 2 (105 ln (5) - \frac{20 5}{3} + \frac{4}{3})

Simplify

= 2 (105 ln (5) + \frac{- 20 5 + 4}{3})

What is the integral from 1 to 5 of sqrt(2x)ln(x^3) ?
The integral from 1 to 5 of sqrt(2x)ln(x^3) is sqrt(2)(10sqrt(5)ln(5)+(-20sqrt(5)+4)/3)