What is ∫ ( t 4 − 2 t 3 − 1 ) d t?

Remove parentheses.∫t4-2t3-1dtSplit the single integral into multiple integrals.∫t4dt+∫-2t3dt+∫-1dtBy the Power Rule, the integral of&nbsp;t4&nbsp;with respect to&nbsp;t&nbsp;is&nbsp;15t5.15t5+C+∫-2t3dt+∫-1dtSince&nbsp;-2&nbsp;is constant with respect to&nbsp;t, move&nbsp;-2&nbsp;out of the integral.15t5+C-2∫t3dt+∫-1dtBy the Power Rule, the integral of&nbsp;t3&nbsp;with respect to&nbsp;t&nbsp;is&nbsp;14t4.15t5+C-2(14t4+C)+∫-1dtApply the constant rule.15t5+C-2(14t4+C)-t+CSimplify.t55-t42-t+CReorder terms.15t5-12t4-t+C

Solved: "What is ∫ ( t 4 − 2..."

Jaylene Duarte

Answered question

2022-04-04

What is

\int (t^{4} - 2 t^{3} - 1) ﻿ d t

Answer & Explanation

Taniya Wood

Beginner2022-04-05Added 19 answers

Remove parentheses.

$\int t^{4} - 2 t^{3} - 1 d t$

Split the single integral into multiple integrals.

$\int t^{4} d t + \int - 2 t^{3} d t + \int - 1 d t$

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

$\frac{1}{5} t^{5} + C + \int - 2 t^{3} d t + \int - 1 d t$

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

$\frac{1}{5} t^{5} + C - 2 \int t^{3} d t + \int - 1 d t$

By the Power Rule, the integral of $t^{3}$ with respect to $t$ is $\frac{1}{4} t^{4}$ .

$\frac{1}{5} t^{5} + C - 2 (\frac{1}{4} t^{4} + C) + \int - 1 d t$

Apply the constant rule.

$\frac{1}{5} t^{5} + C - 2 (\frac{1}{4} t^{4} + C) - t + C$

Simplify.

$\frac{t^{5}}{5} - \frac{t^{4}}{2} - t + C$

Reorder terms.

$\frac{1}{5} t^{5} - \frac{1}{2} t^{4} - t + C$

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