JEE Main Integration Practice Test

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Question: 1 / 400

What is the integral of cos^2(x)?

(1/2)(x + sin(x)cos(x)) + C

To find the integral of $ \cos^2(x) $, one effective method is to use the trigonometric identity that allows us to express $ \cos^2(x) $ in a more manageable form. The relevant identity is:

\cos^2(x) = \frac{1 + \cos(2x)}{2}

Using this identity, we can rewrite the integral as follows:

\int \cos^2(x) \, dx = \int \frac{1 + \cos(2x)}{2} \, dx = \frac{1}{2} \int (1 + \cos(2x)) \, dx

This integral can be separated:

= \frac{1}{2} \left( \int 1 \, dx + \int \cos(2x) \, dx \right)

Evaluating these integrals, we have:

\int 1 \, dx = x

\int \cos(2x) \, dx = \frac{1}{2} \sin(2x)

Putting everything together, we find:

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(1/2)(x + sin^2(x)) + C

(1/2)(x + cos^2(x)) + C

(1/2)(x + tan(x)) + C

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