Find \int \frac{20x^3}{1+x^4} dx.

Category: QuestionsFind \int \frac{20x^3}{1+x^4} dx.
Editor">Editor Staff asked 11 months ago

Find \int \frac{20x^3}{1+x^4} dx.
 
(a) 5 log⁡(x^4)+C
 
(b) -5 log⁡(1+x^4)+C
 
(c) 5 log⁡(1+x^4)+C
 
(d) log⁡(1+x^4)+C
 
I had been asked this question in class test.
 
My question is taken from Methods of Integration-1 in division Integrals of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

Correct answer is (c) 5 log⁡(1+x^4)+C
 
To elaborate: Let 1+x^4=t
 
4x^3 dx=dt
 
∴\int \frac{20x^3}{1+x^4} dx=5\int \frac{dt}{t}
 
=5 log⁡t
 
Replacing t with 1+x^4, we get
 
\int \frac{20x^3}{1+x^4} dx=5 \,log⁡(1+x^4)+C