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CBSE Class 12 Math 2008 Solved Paper

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Question : 6 of 29
Marks: +1, -0
Evaluate: ∫ x21+x3\frac{x^2}{1+x^3} dx
Solution:  
∫ x21+x3\frac{x^2}{1+x^3} dx
Let 1 + x3x^3 = t
⇒ 0 + 3x2dx3x^2dx = dt
⇒ x2dxx^2dx = dt3\frac{dt}{3}
∴ ∫ (x21+x3)\left(\frac{x^2}{1+x^3}\right) dx = ∫ dt3t\frac{\frac{dt}{3}}{t}
= 13∫dtt\frac{1}{3} \int \frac{dt}{t}
= 13\frac{1}{3} log |t| + c
= 13\frac{1}{3} log ∣1+x3∣\left|1+x^3\right| + c
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