Find out the answer to "Evaluate the indefinite integral. ∫12x2+24xx3+3x2+2"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Algotssleeddynf

Answered question

2021-12-29

Evaluate the indefinite integral.

\int \frac{12 x^{2} + 24 x}{x^{3} + 3 x^{2} + 2}

Answer & Explanation

Stella Calderon

Beginner2021-12-30Added 35 answers

Step 1
Consider the provided indefinite integral,
Simplify the given indefinite integral as follows,

\int \frac{12 x^{2} + 24 x}{x^{3} + 3 x^{2} + 2}

Apply u-substotution,
let

u = x^{3} + 3 x^{2} + 2

\frac{d u}{d x} = 3 x^{2} + 6 x

d u = (3 x^{2} + 6 x) d x

multiply 4 in both the sides,

4 d u = (12 x^{2} + 24 x) d x

Step 2
Now, the given indefinite integral is written as,

\int \frac{12 x^{2} + 24 x}{x^{3} + 3 x^{2} + 2} d x = \int \frac{4 d u}{u}

= 4 \int \frac{d u}{u}

= 4 \ln | u | + C

Substitute back,

u = x^{3} + 3 x^{2} + 2

= 4 \ln | x^{3} + 3 x^{2} + 2 | + C

Thus,

\int \frac{12 x^{2} + 24 x}{x^{3} + 3 x^{2} + 2} d x = 4 \ln | x^{3} + 3 x^{2} + 2 | + C

Ella Williams

Beginner2021-12-31Added 28 answers

\int \frac{12 \cdot x^{2} + 24 x}{x^{3} + 3 \cdot x^{2} + 2} d x

Lets

karton

Expert2022-01-04Added 613 answers

$\begin{matrix} \int \frac{12 x^{2} + 24 x}{x^{3} + 3 x^{2} + 2} \\ = 4 \int \frac{1}{u} d u \\ \int \frac{1}{u} d u \\ = \ln (u) \\ 4 \int \frac{1}{u} d u \\ = \ln (u) \\ 4 \int \frac{1}{u} d u \\ = 4 \ln (u) \\ = 4 \ln (x^{3} + 3 x^{2} + 2) \\ \int \frac{12 x^{2} + 24 x}{x^{3} + 3 x^{2} + 2} d x \\ = 4 \ln (| x^{3} + 3 x^{2} + 2 |) + C \end{matrix}$

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