The integration of tan square x $\tan ^2 x$, can be found using trigonometry identities and fundamental integral formulas . The integral of $\tan ^2 x$ with respect to (x) is:

\[

\int \tan^2 x \, dx =\tan x -x + C

\]

Here, (C) represents the constant of integration, which is added because the process of integration determines the antiderivative up to an arbitrary constant.

## Proof of integration of tan square x

We know from trigonometry identities

$\sec^2x =1 + \tan^2x$

or $\tan^2x =\sec^2 x -1$

$\int \tan^2 x dx = \int (\sec^2 x -1) dx$

Now we know that from fundamental integration formula

$\int ( \sec^2 x) \; dx = \tan x + C$

Therefore

$$\int \tan^2 x \; dx = \int (\sec^2 x -1) dx= \tan x -x + C$$

## Definite Integral of tan square x

To find the definite integral of $\tan^2x$ 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.

The definite integral of $\tan^2x$ from $a$ to $b$ is given by:

$$\int_{a}^{b} \tan^2x \, dx = \tan (b) – \tan (a) + a -b $$

This expression represents the accumulated area under the curve of $\tan^2x$ from $x = a$ to $x = b$.

## Solved Examples

**Question 1**

$$ \int \sqrt {tan x} (1 + tan^2) \; dx$$

**Solution**

As We know from trigonometry identities

$\sec^2x =1 + \tan^2x$

$ \int \sqrt {tan x} (1 + tan^2) \; dx=\int \sqrt {tan x} \sec^2 x $

Now let u=tan x

then $ du = \sec^2 x dx$, So we get

$=\int t^{1/2} \; dt $

$=\frac {2}{3} t^{3/2} + C$

Substituting back the values

$=\frac {2}{3} (\tan x) ^{3/2} + C$

**Question 2**

\[

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

\]

**Solution**

we can use a trigonometric identity to simplify the expression. The key identity to use here is

$$

1 + \tan^2(x) = \sec^2(x)

$$

which means the integral simplifies to

$$

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

$$

To solve (\int \cos^2(x) \, dx), we can use the half-angle identity:

$$

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

$$

So, we get

$$

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

$$

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