"Find the integral, full solution ∫(2x+3)2dx" was answered by our community

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Monique Slaughter

Answered question

2022-01-04

Find the integral, full solution

\int {(2 x + 3)}^{2} d x

Answer & Explanation

Mary Herrera

Beginner2022-01-05Added 37 answers

Step 1
Given integral,

\int {(2 x + 3)}^{2} d x

Step 2

\int {(2 x + 3)}^{2} d x = \int (4 x^{2} + 12 x + 9) d x

= \int 4 x^{2} d x + \int 12 x d x + \int 9 d x

= 4 \int x^{2} d x + 12 \int x d x + 9 \int d x

= 4 \frac{x^{3}}{3} + 12 \frac{x^{2}}{2} + 9 x + C

Since

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C, n \neq 1

= \frac{4 x^{3}}{3} + 6 x^{2} + 9 x + C

Raymond Foley

Beginner2022-01-06Added 39 answers

Calculate:

\int {(2 x + 3)}^{2} d x

= \frac{1}{2} \int u^{2} d u

\int u^{2} d u

= \frac{u^{3}}{3}

\frac{1}{2} \int u^{2} d u

= \frac{u^{3}}{6}

= \frac{{(2 x + 3)}^{3}}{6}

= \frac{{(2 x + 3)}^{3}}{6} + C

Lets

karton

Expert2022-01-11Added 613 answers

$\int (2 x + 3)^{2} d x \int 4 x^{2} + 12 x + 9 d x \int 4 x^{2} d x + \int 12 x d x + \int 9 d x \frac{4 x^{3}}{3} + 6 x^{2} + 9 x Add C Answer: \frac{4 x^{3}}{3} + 6 x^{2} + 9 x + C$

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