First slide

Introduction to integration

Question

If $\int \frac{x + 1}{(x^{2} + x + 1)} \frac{d x}{\sqrt{x^{2} + x + 1}} = k \frac{x - 1}{\sqrt{x^{2} + x + 1}} + C$ then the value of $k$

Easy

A

1/2
B

1/3
C

2/3
D

none of these

Solution

Differentiating both the sides w.r.t.x, and simplifying , we get
$\frac{x + 1}{{(x^{2} + x + 1)}^{3 / 2}} = \frac{3 k}{2} \frac{x + 1}{{(x^{2} + x + 1)}^{3 / 2}} \Rightarrow k = \frac{2}{3}$

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