Integrate 7 8 t 4 ln ⁡ ( t ) 2 with respect to t.

Rewrite as&nbsp;∫7t4(2ln(t))8dt.∫7t4(2ln(t))8dtSince&nbsp;7⋅28&nbsp;is constant with respect to&nbsp;t, move&nbsp;7⋅28&nbsp;out of the integral.7⋅28∫t4(ln(t))dtSimplify.74∫t4(ln(t))dtIntegrate by parts using the formula&nbsp;∫udv=uv-∫vdu, where&nbsp;u=ln(t)&nbsp;and&nbsp;dv=t4.74(ln(t)(15t5)-∫15t51tdt)Simplify.74(ln(t)t55-∫t45dt)Since&nbsp;15&nbsp;is constant with respect to&nbsp;t, move&nbsp;15&nbsp;out of the integral.74(ln(t)t55-(15∫t4dt))By the Power Rule, the integral of&nbsp;t4&nbsp;with respect to&nbsp;t&nbsp;is&nbsp;15t5.74(ln(t)t55-15(15t5+C))Simplify the answer.74(15ln(t)t5-125t5)+C&nbsp;

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Berghofaei0e

Answered question

2022-05-11

Integrate

\frac{7}{8} ﻿ t^{4} \ln (t)^{2}

with respect to t.

Answer & Explanation

Makhi Lyons

Beginner2022-05-12Added 15 answers

Rewrite as $\int \frac{7 t^{4} (2 \ln (t))}{8} d t$ .

$\int \frac{7 t^{4} (2 \ln (t))}{8} d t$

Since $\frac{7 \cdot 2}{8}$ is constant with respect to $t$ , move $\frac{7 \cdot 2}{8}$ out of the integral.

$\frac{7 \cdot 2}{8} \int t^{4} (\ln (t)) d t$

Simplify.

$\frac{7}{4} \int t^{4} (\ln (t)) d t$

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (t)$ and $d v = t^{4}$ .

$\frac{7}{4} (\ln (t) (\frac{1}{5} t^{5}) - \int \frac{1}{5} t^{5} \frac{1}{t} d t)$

Simplify.

$\frac{7}{4} (\frac{\ln (t) t^{5}}{5} - \int \frac{t^{4}}{5} d t)$

Since $\frac{1}{5}$ is constant with respect to $t$ , move $\frac{1}{5}$ out of the integral.

$\frac{7}{4} (\frac{\ln (t) t^{5}}{5} - (\frac{1}{5} \int t^{4} d t))$

By the Power Rule, the integral of $t^{4}$ with respect to $t$ is $\frac{1}{5} t^{5}$ .

$\frac{7}{4} (\frac{\ln (t) t^{5}}{5} - \frac{1}{5} (\frac{1}{5} t^{5} + C))$

Simplify the answer.

$\frac{7}{4} (\frac{1}{5} \ln (t) t^{5} - \frac{1}{25} t^{5}) + C$

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