Evaluate ∫ 6 t 7 3 t + 2 3 d t.

Question

Evaluate   ∫            6      t              7                        3          t          +          2                3                  ﻿    d  t.

graffus1hb30 · Accepted Answer

This integral could not be completed using integration by parts. Mathway will use another method.

Since $\frac{6}{7}$ is constant with respect to $t$ , move $\frac{6}{7}$ out of the integral.

$\frac{6}{7} \int \frac{t}{\sqrt[3]{3 t + 2}} d t$

Let $u = 3 t + 2$ . Then $d u = 3 d t$ , so $\frac{1}{3} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

$\frac{6}{7} \int \frac{\frac{u}{3} - \frac{2}{3}}{\sqrt[3]{u}} \cdot \frac{1}{3} d u$

Simplify.

$\frac{6}{7} \int \frac{u - 2}{9 \sqrt[3]{u}} d u$

Since $\frac{1}{9}$ is constant with respect to $u$ , move $\frac{1}{9}$ out of the integral.

$\frac{6}{7} (\frac{1}{9} \int \frac{u - 2}{\sqrt[3]{u}} d u)$

Simplify the expression.

$\frac{2}{21} \int (u - 2) u^{- \frac{1}{3}} d u$

Expand $(u - 2) u^{- \frac{1}{3}}$ .

$\frac{2}{21} \int u^{\frac{2}{3}} - 2 u^{- \frac{1}{3}} d u$

Split the single integral into multiple integrals.

$\frac{2}{21} (\int u^{\frac{2}{3}} d u + \int - 2 u^{- \frac{1}{3}} d u)$

By the Power Rule, the integral of $u^{\frac{2}{3}}$ with respect to $u$ is $\frac{3}{5} u^{\frac{5}{3}}$ .

$\frac{2}{21} (\frac{3}{5} u^{\frac{5}{3}} + C + \int - 2 u^{- \frac{1}{3}} d u)$

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

$\frac{2}{21} (\frac{3}{5} u^{\frac{5}{3}} + C - 2 \int u^{- \frac{1}{3}} d u)$

By the Power Rule, the integral of $u^{- \frac{1}{3}}$ with respect to $u$ is $\frac{3}{2} u^{\frac{2}{3}}$ .

$\frac{2}{21} (\frac{3}{5} u^{\frac{5}{3}} + C - 2 (\frac{3}{2} u^{\frac{2}{3}} + C))$

Simplify.

$\frac{2}{21} (\frac{3 u^{\frac{5}{3}}}{5} - 3 u^{\frac{2}{3}}) + C$

Replace all occurrences of $u$ with $3 t + 2$ .

$\frac{2}{21} (\frac{3 {(3 t + 2)}^{\frac{5}{3}}}{5} - 3 {(3 t + 2)}^{\frac{2}{3}}) + C$

Simplify.

$\frac{6 {(3 t + 2)}^{\frac{2}{3}} (t - 1)}{35} + C$

Evaluate <mrow> 6 t </mrow> <mrow> 7 <mroot> <mro

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