Question

# Find all the antiderivatives of the following function. Check your work by taking derivatives.p(x)=3\sec^{2}xZKS

Derivatives

Find all the antiderivatives of the following function. Check your work by taking derivatives.
$$p(x)=3\sec^{2}x$$

2021-02-12
Step 1
To find an antiderivative of the function: $$\displaystyle{p}{\left({x}\right)}={3}{{\sec}^{{{2}}}{x}}$$
Solution:
Given function is: $$\displaystyle{p}{\left({x}\right)}={3}{{\sec}^{{{2}}}{x}}$$.
For finding antiderivative we need to integrate the given function.
$$\displaystyle\int{3}{{\sec}^{{{2}}}{x}}{\left.{d}{x}\right.}={3}\int{{\sec}^{{{2}}}{x}}{\left.{d}{x}\right.}$$
$$\displaystyle={3}{\tan{{x}}}+{c}$$ (using $$\displaystyle\int{{\sec}^{{{2}}}{x}}{\left.{d}{x}\right.}={\tan{{x}}}+{c}$$)
Verification:
Let,
$$\displaystyle{y}={3}{\tan{{x}}}+{c}$$
differentiating both sides w.r.t x we get:
$$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({3}{\tan{{x}}}+{c}\right)}$$
$$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={3}{\left({{\sec}^{{{2}}}{x}}\right)}+{0}$$ (using, $$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\tan{{x}}}\right)}={{\sec}^{{{2}}}{x}},{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{n}}}\right)}={n}{x}^{{{n}-{1}}}$$)
$$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={3}{{\sec}^{{{2}}}{x}}$$, which is given function, $$\displaystyle{p}{\left({x}\right)}={3}{{\sec}^{{{2}}}{x}}$$
hence, veryfied
Step 2
Result:
$$\displaystyle\int{3}{{\sec}^{{{2}}}{x}}{\left.{d}{x}\right.}={3}{\tan{{x}}}+{c}$$