Question

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

Derivatives
ANSWERED
asked 2021-02-10

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

Answers (1)

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:
\(\displaystyle\int{3}{{\sec}^{{{2}}}{x}}{\left.{d}{x}\right.}={3}{\tan{{x}}}+{c}\)
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