1
$\begingroup$

How do I integrate $\dfrac{1}{(\sin x + \cos x)^{4}}$? I could not think of any way. I tried substitution but of no use.

$\endgroup$
  • $\begingroup$ use the $\sin x + \cos x =\sqrt 2 \sin(\pi/4+x)$ $\endgroup$ – Math-fun Aug 10 '15 at 19:27
  • $\begingroup$ i would use the tan-half angle substitution $\endgroup$ – Dr. Sonnhard Graubner Aug 10 '15 at 19:28
5
$\begingroup$

$$\int \frac{1}{(\sin x + \cos x)^{4}}dx$$

Hint:

Multiply numerator and denominator by $\sec^4$

$$=\int \frac{\sec^4 (x)} { 1+4\tan(x)+6\tan^2 (x)+4\tan^3(x)+\tan^4(x)}dx$$

Use $\sec^2(x)=\tan^2(x)+1$

$$\int \frac{(1+\tan^2(x))\sec^2(x)}{(1+\tan(x))^4}dx$$

Now substitute $u=\tan(x)$

$$\int \frac{u^2+1}{(u+1)^4}du$$

I hope that you can finish from here.

$\endgroup$
  • 1
    $\begingroup$ It might be easier to leave the denominator as $(1+\tan x)^4$ and substitute $u = 1+\tan x$. $\endgroup$ – JimmyK4542 Aug 10 '15 at 19:29
  • $\begingroup$ @Nehorai Yes Thanks $\endgroup$ – Taylor Ted Aug 10 '15 at 19:35
4
$\begingroup$

$$\sin x + \cos x = \sqrt{2} \left( \sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4} \right) = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right).$$

$\endgroup$
  • $\begingroup$ what should i do next for integration, next step? $\endgroup$ – Taylor Ted Aug 11 '15 at 12:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.