Determine the following indefinite integral. \int \frac{dx}{1-\sin^{2}x}

Jerold 2021-10-29 Answered
Determine the following indefinite integral.
\(\displaystyle\int{\frac{{{\left.{d}{x}\right.}}}{{{1}-{{\sin}^{{{2}}}{x}}}}}\)

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Expert Answer

dieseisB
Answered 2021-10-30 Author has 12638 answers
Step 1
Given integral:
\(\displaystyle\int{\frac{{{\left.{d}{x}\right.}}}{{{1}-{{\sin}^{{{2}}}{x}}}}}\)
Step 2
Now,
Use the following identity: \(\displaystyle{1}-{{\sin}^{{{2}}}{\left({x}\right)}}={{\cos}^{{{2}}}{\left({x}\right)}}\)
\(\displaystyle\int{\frac{{{1}}}{{{1}-{{\sin}^{{{2}}}{\left({x}\right)}}}}}{\left.{d}{x}\right.}=\int{\frac{{{1}}}{{{{\cos}^{{{2}}}{\left({x}\right)}}}}}{\left.{d}{x}\right.}\)
Use the following identity: \(\displaystyle{\frac{{{1}}}{{{\cos{{\left({x}\right)}}}}}}={\sec{{\left({x}\right)}}}\)
\(\displaystyle=\int{{\sec}^{{{2}}}{\left({x}\right)}}{\left.{d}{x}\right.}\)
Use the common integral: \(\displaystyle\int{{\sec}^{{{2}}}{\left({x}\right)}}{\left.{d}{x}\right.}={\tan{{\left({x}\right)}}}\)
\(\displaystyle={\tan{{\left({x}\right)}}}\)
Add a constant to the solution:
\(\displaystyle={\tan{{\left({x}\right)}}}+{C}\)
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