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# Integration of cos square x

Integration of cos square x can be calculated using trigonometric identities .Here is the formula for it

$$\int \cos^2(x) \, dx = \frac{1}{2} x + \frac{1}{4} \sin(2x) + C$$

## Proof of Integration of cos square x

To integrate $\cos^2(x)$, we use a trigonometric identity to simplify the expression before integrating. The identity commonly used is the power-reduction formula or double angle identity

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

Now, let’s integrate using this identity:

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

This integral can be split into two simpler integrals:

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

Now, integrate each part:

1. The integral of 1 with respect to $x$ is $x$.
2. The integral of $\cos(2x)$ is $\frac{\sin(2x)}{2}$.

Proof of this Integral $\int \cos(2x) dx= \frac{\sin(2x)}{2}$

Let t=2x
dt=2 dx
or
dx= dt/2
Therefore
$\int \cos(2x) \; dx== \frac {1}{2} \int cos t \; dt= \frac{\sin(2x)}{2}$

So, the integral becomes:

$$= \frac{1}{2} x + \frac{1}{4} \sin(2x) + C$$

where $C$ is the constant of integration. Therefore, the integral of $\cos^2(x)$ is:

$$\int \cos^2(x) \, dx = \frac{1}{2} x + \frac{1}{4} \sin(2x) + C$$

## Definite Integral of cos square x

To find the definite integral of $\cos^2(x)$ over a specific interval, we use the same approach as with the indefinite integral, but we’ll apply the limits of integration at the end.

So, the definite integral of $\cos^2(x)$ from ( a ) to ( b ) is:

$\int_a^b \cos^2(x) \, dx = \frac{1}{2} (b – a) + \frac{1}{4} (\sin(2b) – \sin(2a))$

Example

$$\int_0^\pi \cos^2(x) \, dx$$

First, we use the power-reduction formula:

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

Now, integrate over the interval $[0, \pi]$:

$$\int_0^\pi \frac{1 + \cos(2x)}{2} \, dx$$

$$= \frac{1}{2} \int_0^\pi dx + \frac{1}{2} \int_0^\pi \cos(2x) \, dx$$

$$= \frac{1}{2} x \Big|_0^\pi + \frac{1}{4} \sin(2x) \Big|_0^\pi$$
=\frac{\pi}{2} + 0=\frac{\pi}{2}

Therefore, the definite integral of $\cos^2(x)$ from $0$ to $\pi$ is $\frac{\pi}{2}$. This result represents the area under the curve of $\cos^2(x)$ between $x = 0$ and $x = \pi$.

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