What is the integral from 2 to 5 of t^3ln(5t) ?

The integral from 2 to 5 of t^3ln(5t) is -4ln(10)-609/16+625/2 ln(5)

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integral from 2 to 5 of t^3ln(5t)

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

- 4 ln (10) - \frac{609}{16} + \frac{625}{2} ln (5)

Decimal Notation

455.67650 \dots

Solution steps

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

Apply Integration By Parts

: [\frac{1}{4} t^{4} ln (5 t) - \int \frac{t ^{3}}{4} d t]_{2}^{5}

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

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

= [\frac{1}{4} t^{4} ln (5 t) - \frac{t ^{4}}{16}]_{2}^{5}

Compute the boundaries:

- 4 ln (10) - \frac{609}{16} + \frac{625}{2} ln (5)

= - 4 ln (10) - \frac{609}{16} + \frac{625}{2} ln (5)

\frac{d}{d x} ((x - 1) (x - 2) (2 - x^{2}))

\frac{3}{6 t + 3}

f^{^{'}} (t) = \frac{8 t}{8 + t}

f (x) = (2 x^{2} - 1)^{\frac{1}{2}}

y^{^{''}} + 8 y^{^{'}} + 65 y = 0, y (0) = 0, y^{^{'}} (0) = 0

What is the integral from 2 to 5 of t^3ln(5t) ?
The integral from 2 to 5 of t^3ln(5t) is -4ln(10)-609/16+625/2 ln(5)