Question

What is the antiderivative of $${\sec }^2(x)$$?

Answer 1

Hint: From the question given we have to find the antiderivative of $${\sec }^2(x)$$. Generally, antiderivatives are opposite to the derivatives (inverse derivatives). We know that the derivative of$$\tan (x)$$ is $${\sec }^2(x)$$
We need to find the antiderivative of $${\sec }^2(x)$$. Antiderivative means integral. From this we will get the antiderivative of $${\sec }^2(x)$$.

Complete step by step solution:
Generally, antiderivatives are opposite to the derivatives (inverse derivatives).
We know that the derivative of$$\tan (x)$$ is $${\sec }^2(x)$$
We need to find the antiderivative of $${\sec }^2(x)$$.
$$\Rightarrow$$$${\displaystyle \frac{d}{dx}}(\tan x)={\sec }^2(x)$$
From the above equation it is clear that the derivative of $$\tan (x)$$ is $${\sec }^2(x)$$.
We know that the antiderivatives are inverse derivatives of the derivatives.
So, it is very clear that the antiderivative of the $${\sec }^2(x)$$ becomes$$\tan (x)$$.
$$\Rightarrow$$$${\displaystyle \frac{d}{dx}}(\tan x)={\sec }^2(x)$$
$$\Rightarrow$$$$\tan x=\int {\sec }^2(x)+c$$.
Integral is nothing but the antiderivative.
$$\Rightarrow \int {\sec }^{{}^2}(x)=\tan x+c$$
Here c is some constant value.
So, the antiderivative of the $${\sec }^2(x)$$ becomes$$\tan (x)$$.
Antiderivative of $${\sec }^2(x)$$ is$$\tan (x)$$+c.
Antiderivative of $${\sec }^2(x)$$= $$\tan (x)$$+c
$$\Rightarrow$$$${\displaystyle \frac{d}{dx}}(\tan x)={\sec }^2(x)$$
$$\Rightarrow$$ $$\int {\sec }^{{}^2}(x)=\tan (x)+c$$
Antiderivative is $$\tan (x)$$+c.

So, the antiderivative of $${\sec }^2(x)$$ is $$\tan (x)$$+c.

Note: Students must know the basis derivatives of trigonometric functions like:
$$\Rightarrow {\displaystyle \frac{d}{dx}}(\tan x)={\sec }^2(x)$$
$$\Rightarrow {\displaystyle \frac{d}{dx}}(\sin x)=\cos x$$
$$\Rightarrow {\displaystyle \frac{d}{dx}}(\cos x)=-\sin x$$
Students must know the concept of antiderivative. Students must be very careful while doing the calculations.