What is the integral from 1 to 3 of 3r^3ln(r) ?

The integral from 1 to 3 of 3r^3ln(r) is 3(-5+81/4 ln(3))

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integral from 1 to 3 of 3r^3ln(r)

\int_{1}^{3} 3 r^{3} ln (r) d r

3 (- 5 + \frac{81}{4} ln (3))

Decimal Notation

51.74069 \dots

Solution steps

\int_{1}^{3} 3 r^{3} ln (r) d r

Take the constant out:

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

= 3 \cdot \int_{1}^{3} r^{3} ln (r) d r

Apply Integration By Parts

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

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

\int \frac{r ^{3}}{4} d r = \frac{r ^{4}}{16}

= 3 [\frac{1}{4} r^{4} ln (r) - \frac{r ^{4}}{16}]_{1}^{3}

Compute the boundaries:

- 5 + \frac{81}{4} ln (3)

= 3 (- 5 + \frac{81}{4} ln (3))

\int (7 x - 11)^{4} d x

y^{^{''}} - 3 y^{^{'}} + 2 y = 2 t^{3}

\frac{d}{d x} (ln (x^{4} - 6 x^{2}))

\int tan^{3} (21 π x) d x

g (x) = \frac{sin ( 8 x )}{cos ( x )}

What is the integral from 1 to 3 of 3r^3ln(r) ?
The integral from 1 to 3 of 3r^3ln(r) is 3(-5+81/4 ln(3))