Algotssleeddynf

2021-12-29

Evaluate the indefinite integral.

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

Stella Calderon

Beginner2021-12-30Added 35 answers

Step 1

Consider the provided indefinite integral,

Simplify the given indefinite integral as follows,

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

Apply u-substotution,

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

$\frac{du}{dx}=3{x}^{2}+6x$

$du=(3{x}^{2}+6x)dx$

multiply 4 in both the sides,

$4du=(12{x}^{2}+24x)dx$

Step 2

Now, the given indefinite integral is written as,

$\int \frac{12{x}^{2}+24x}{{x}^{3}+3{x}^{2}+2}dx=\int \frac{4du}{u}$

$=4\int \frac{du}{u}$

$=4\mathrm{ln}\left|u\right|+C$

Substitute back,$u={x}^{3}+3{x}^{2}+2$

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

Thus,$\int \frac{12{x}^{2}+24x}{{x}^{3}+3{x}^{2}+2}dx=4\mathrm{ln}|{x}^{3}+3{x}^{2}+2|+C$

Consider the provided indefinite integral,

Simplify the given indefinite integral as follows,

Apply u-substotution,

let

multiply 4 in both the sides,

Step 2

Now, the given indefinite integral is written as,

Substitute back,

Thus,

Ella Williams

Beginner2021-12-31Added 28 answers

Lets

karton

Expert2022-01-04Added 613 answers

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