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How can we find the indefinite integral for:

$$\int \frac{\sqrt{\cot(x)} - \sqrt{\tan(x)}}{4+3 \sin^2 (x)} \ \mathrm d x$$

I tried expressing the denominator in $\sin x+\cos x$ form since we have its derivative in numerator but I am not able to proceed. Please help me out.

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    $\begingroup$ How to use MathJax. Where is this problem from? $\endgroup$
    – Toby Mak
    Commented Sep 7, 2020 at 9:03
  • $\begingroup$ It is from my Calculus Course book . I am in 12th Grade so I just know some basic integration $\endgroup$ Commented Sep 7, 2020 at 9:15
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    $\begingroup$ @ZAhmed I believe the combination of the hypergeometric functions in Mathematica simplify to an elementary expression. $\endgroup$ Commented Sep 7, 2020 at 9:39

1 Answer 1

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By some basic trigonometric identities, one has $$\int \frac{\sqrt{\cot(x)} - \sqrt{\tan(x)}}{4+3 \sin^2 (x)}~dx=\int \frac{\sec^2(x)(1-\tan(x))}{\sqrt{\tan(x)}(7\tan^2(x)+4)}~dx,$$ which motivates the substitution $u=\tan(x)$. This leads to $$\int \frac{1-u}{\sqrt{u}(7u^2+4)}~du,$$ whose integrand can be converted to a rational expression via the substitution $u=v^2$. The resulting expression can be solved using partial fractions.

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  • $\begingroup$ Thank you for your solution :) @ projectilemotion $\endgroup$ Commented Sep 7, 2020 at 9:43

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