Question:
(c) Find the value of \( \int \sqrt{5 + 2x + x^2} \, dx \).
(c) Find the value of \( \int \sqrt{5 + 2x + x^2} \, dx \).
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Complete the square before integrating expressions under a square root.
Updated On: Mar 1, 2025
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Solution and Explanation
Completethesquareintheexpression\(5+2x+x^2\):
\[
5+2x+x^2=(x+1)^2+4.
\]
Substitute\(u=x+1\),then\(du=dx\):
\[
\int\sqrt{5+2x+x^2}\,dx=\int\sqrt{u^2+4}\,du.
\]
Solveusingtrigonometricsubstitution:
\[
u=2\tan\theta\quad\Rightarrow\quad\sqrt{u^2+4}=2\sec\theta.
\]
Evaluateandsimplifytheintegral.
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