Question

The integral $16 \int\limits_1^2 \frac{d x}{x^3\left(x^2+2\right)^2}$ is equal to

• A. $\frac{11}{6}-\log _e 4$
• B. $\frac{11}{12}+\log _{ e } 4$
• C. $\frac{11}{6}+\log _{ e } 4$
• D. $\frac{11}{12}-\log _e 4$

Answer 1

Answer: (C)

$\frac{11}{6}+\log _{ e } 4$

Answer 2

The correct answer is (C) : $\frac{11}{6}+\log _{ e } 4$
I=161∫2​x3(x2+2)2dx​
=161∫2​x3x4(1+x22​)2dx​
Let, 1+x22​=t⇒x3−4​dx=dt
I=−43∫23​​(t−12​)2t2dt​
I=−43∫23​​(2t−1​)2t2dt​
I=−44​3∫23​​(1−t2​+t21​)dt
I=−1[t−2ℓn∣t∣−t1​]323​​
I=−1[(23​−2ℓn23​−32​)−(3−2ℓn3−31​)]
I=−1[2ℓn2−611​]
I=611​−ℓn4