What is the Integration of tan^2x | Integral of tan^2 x

The integration of tan^2x is equal to ∫tan²x dx = tan (x)-x. In this post, we will learn how to find the integral of tan square x. We will use the following two formulas:

1. ∫sec²x dx =tan x+C
2. ∫dx = x+C

Integration of tan²x

Question: What is the integration of tan² x? That is, find ∫tan² x dx.

Answer: ∫tan²x dx = tan(x) -x +C.

Explanation:

To find the integration of tan²x, we will proceed as follows. At first, we will use the formula given below.

sec² x =1+tan² x

⇒ tan² x = sec² x -1

Therefore, we have that ∫tan²x dx = ∫(sec² x-1) dx

= ∫sec² x dx – ∫dx

= tan x – x +C by the above formulas.

So the integration of tan² x is equal to tan(x)-x+C where C is an integration constant.

Definite Integral of tan²x

Question: Find the definite integral $\int_0^{\frac{\pi}{4}} \tan^2 x dx$

Answer:

From the above, we know that the integration of tan square x is ∫tan²x dx = tan x – x +C. Therefore, the given definite integral will be equal to

$\int_0^{\frac{\pi}{4}} \tan^2 x dx$

= $[\tan x -x]_0^{\frac{\pi}{4}}$

= $(\tan \frac{\pi}{4}-  \frac{\pi}{4})$ $-(\tan 0 -0)$

= $(1-  \frac{\pi}{4})-(0 -0)$ as the value of $\tan \frac{\pi}{4}$ is $1$ and the value of tan0 is 0.

= $1-  \frac{\pi}{4}$.

Thus, the integral of tan²x from 0 to π/4 is equal to 1-π/4.

Also Read: 

Integration of log(sinx) from 0 to pi/2

Integration of e^3x

Derivative & Integration of 1/root(x)

Integration of 1/(1+x²)

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