What is {dx}{²(^{-1}x)} equal to?

Correct Answer is : tan⁡−1x+c^{-1} x + ctan−1x+c

Correct Answer is : tan⁡−1x+c^{-1} x + ctan−1x+c

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UPSC NDA I 2021 Mathematics Solved Paper

Question : 101 of 120

Marks: +1, -0

What is

\int\frac{dx}{\sec^{2}(\tan^{-1}x)}

equal to?

Solution:

Concept:

Use the identity

\sec^2 \theta - \tan^2 \theta = 1

to simplify

\sec^2(\tan^{-1}x)

into

1 + x^2

, then integrate.

Explanation:

Let

I = \int\frac{dx}{\sec^{2}(\tan^{-1}x)}

We know

\sec^2\theta = 1 + \tan^2\theta

\sec^{2}(\tan^{-1}x) = 1 + \tan^{2}(\tan^{-1}x) = 1 + x^{2}

Hence

I = \int\frac{dx}{1+x^{2}}

The standard integral

\int\frac{dx}{1+x^{2}} = \tan^{-1}x + C

Thus

I = \tan^{-1}x + C

Alternate substitution: set

\tan^{-1}x = \theta

, then

x = \tan\theta

dx = \sec^2\theta\,d\theta

Then

I = \int\frac{\sec^2\theta\,d\theta}{\sec^2\theta} = \int d\theta = \theta + C = \tan^{-1}x + C

Answer:

\tan^{-1} x + C

, which is option B.

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