What is the integral from 1/3 to 7 of 10xln(3x) ?

The integral from 1/3 to 7 of 10xln(3x) is 10(-110/9+49/2 ln(21))

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integral from 1/3 to 7 of 10xln(3x)

\int_{\frac{1}{3}}^{7} 10 x ln (3 x) d x

10 (- \frac{110}{9} + \frac{49}{2} ln (21))

Decimal Notation

623.68577 \dots

Solution steps

\int_{\frac{1}{3}}^{7} 10 x ln (3 x) d x

Take the constant out:

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

= 10 \cdot \int_{\frac{1}{3}}^{7} x ln (3 x) d x

Apply Integration By Parts

: [\frac{1}{2} x^{2} ln (3 x) - \int \frac{x}{2} d x]_{\frac{1}{3}}^{7}

= 10 [\frac{1}{2} x^{2} ln (3 x) - \int \frac{x}{2} d x]_{\frac{1}{3}}^{7}

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

= 10 [\frac{1}{2} x^{2} ln (3 x) - \frac{x ^{2}}{4}]_{\frac{1}{3}}^{7}

Compute the boundaries:

- \frac{110}{9} + \frac{49}{2} ln (21)

= 10 (- \frac{110}{9} + \frac{49}{2} ln (21))

\int_{- 9}^{- 7} \frac{1}{x} d x

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\int 4 x sin (2 x) d x

(- 3, - 4), (0, 2)

What is the integral from 1/3 to 7 of 10xln(3x) ?
The integral from 1/3 to 7 of 10xln(3x) is 10(-110/9+49/2 ln(21))