Question: "Evaluate the following integral. ∫3x2−2x3−2x−8dx"

hrostentsp6

Answered question

2021-11-16

Evaluate the following integral.

\int \frac{3 x^{2} - 2}{x^{3} - 2 x - 8} d x

Archie Griffin

Beginner2021-11-17Added 9 answers

Step 1
To evaluate the integral

\int \frac{3 x^{2} - 2}{x^{3} - 2 x - 8} d x

Solution:
The given integral is

\int \frac{3 x^{2} - 2}{x^{3} - 2 x - 8} d x

using substitution method as,
Put

x^{3} - 2 x - 8 = t

, we get

(3 x^{2} - 2) d x = d t

Now the integral becomes,
Step 2

\int \frac{d t}{t} = \log t + c

= \log (x^{3} - 2 x - 8) + c

Hence, the value of the integral is

\log (x^{3} - 2 x - 8) + c

Alfonso Miller

Beginner2021-11-18Added 20 answers

Step 1: Use Integration by Substitution.
Let

u = x^{3} - 2 x - 8, d u = 3 x^{2} - 2 d x

Step 2: Using u and du above, rewrite

\int \frac{3 x^{2} - 2}{x^{3} - 2 x - 8} d x

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

Step 3: The derivative of

\ln x i s \frac{1}{x}

\ln u

Step 4: Substitute

u = x^{3} - 2 x - 8

back into the original integral.

\ln (x^{3} - 2 x - 8)

Step 5: Add constant.

\ln (x^{3} - 2 x - 8) + C

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