"Find the indefinite integral. ∫x2x3+1dx" solved and explained

Maggenifh

Answered question

2021-11-20

Find the indefinite integral.

\int \frac{x^{2}}{x^{3} + 1} d x

Answer & Explanation

Coldst

Beginner2021-11-21Added 18 answers

Step 1
To Find the indefinite integral.

I = \int \frac{x^{2}}{x^{3} + 1} d x

Substitution

u = x^{3}

d u = 3 x^{2} d x

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

Step 2
Now,

I = \int \frac{1}{u + 1} (\frac{d u}{3})

I = \frac{1}{3} \int \frac{d u}{u + 1}

I = \frac{1}{3} [\log (u + 1)] + c

I = \frac{1}{3} [\log (x^{3} + 1)] + c

Vincent Diaz

Beginner2021-11-22Added 19 answers

Step 1: Use Integration by Substitution.
Let $u = x^{3} + 1, d u = 3 x^{2} d x, t h e n x^{2} d x = \frac{1}{3} d u$
Step 2: Using u and du above, rewrite $\int \frac{x^{2}}{x^{3} + 1} d x$ .
$\int \frac{1}{3 u} d u$
Step 3: Use Constant Factor Rule: $\int c f (x) d x = c \int f (x) d x$ .
$\frac{1}{3} \int \frac{1}{u} d u$
Step 4: The derivative of $\ln x i s \frac{1}{x}$ .
$\frac{\ln u}{3}$
Step 5: Substitute $u = x^{3} + 1$ back into the original integral.
$\frac{\ln (x^{3} + 1)}{3}$
Step 6: Add constant.
$\frac{\ln (x^{3} + 1)}{3} + C$

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