2
$\begingroup$

Evaluation of $\displaystyle \int\frac{\sin (x+\alpha)}{\cos^3 x}\cdot \sqrt{\frac{\csc x+\sec x}{\csc x-\sec x}}dx$

$\bf{My \; Try::}$ Let $\displaystyle I = \int \frac{\sin (x+\alpha)}{\cos^3 x}\cdot \sqrt{\frac{\cos x+\sin x}{\cos x-\sin x}}dx$

So $\displaystyle I = \int\frac{\sin (x+\alpha)}{\cos x}\cdot \sqrt{\frac{\cos x+\sin x}{\cos x-\sin x}}dx\cdot \sec^2 x$

$\displaystyle I = \int \frac{\sin x\cdot \cos \alpha+\cos x\cdot \sin \alpha}{\cos x}\cdot \sqrt{\frac{\cos +\sin x}{\cos x-\sin x}}\cdot \sec^2 x dx$

$\displaystyle I = \int\left(\cos \alpha\cdot \tan x+\sin \alpha\right)\cdot \sqrt{\frac{1+\tan x}{1-\tan x}}\cdot \sec^2 xdx$

Now Let $\tan x= t\;,$ Then $\sec^2 xdx = dt$

$\displaystyle I = \int (\cos \alpha\cdot t + \sin \alpha)\sqrt{\frac{1+t}{1-t}}dt$

Now How can I solve after that

Help me

Thanks

$\endgroup$
1
$\begingroup$

Let \begin{align} I = \int \frac{\sin(x+a)}{\cos^3(x)}\cdot \sqrt{\frac{\cos x+\sin x}{\cos x-\sin x}}dx \end{align} then \begin{align} I &= \int (\sin(x) \cos(a) + \cos(x) \sin(a))(\cos(x) + \sin(x)) \ \frac{dx}{\cos^{3}(x) \sqrt{\cos(2x)}} \\ &= \cos(a) \ \int \frac{\sin^{2}(x) \ dx}{\cos^{3}(x) \sqrt{\cos(2x)}} + \sin(a) \ \int \frac{ dx}{\cos(x) \sqrt{\cos(2x)}} + (\cos(a) + \sin(a)) \ \int \frac{\sin(x) \ dx}{\cos^{2}(x) \sqrt{\cos(2x)}} \\ &= \frac{1}{2} (\cos(a) + 2 \sin(a)) \ \tan^{-1}\left( \frac{\sin(x)}{\sqrt{\cos(2x)}}\right) - \frac{1}{2} \cos(a) \sqrt{\cos(2x)} \ \frac{\sin(x)}{\cos^{2}(x)} - (\cos(a) + \sin(a)) \ \frac{\sqrt{\cos(2x)}}{\cos(x)} \end{align} where there integrals \begin{align} \int \frac{\sin^{2}(x) \ dx}{\cos^{3}(x) \sqrt{\cos(2x)}} &= \frac{1}{2} \left[ \tan^{-1}\left( \frac{\sin(x)}{\sqrt{\cos(2x)}}\right) - \frac{\sin(x) \ \sqrt{\cos(2x)}}{\cos^{2}(x)} \right] \\ \int \frac{\sin(x) \ dx}{\cos^{2}(x) \sqrt{\cos(2x)}} &= - \frac{ \sqrt{\cos(2x)}}{\cos^{2}(x)} \\ \int \frac{ dx}{\cos(x) \sqrt{\cos(2x)}} &= \tan^{-1}\left( \frac{\sin(x)}{\sqrt{\cos(2x)}}\right) \end{align} were used.

$\endgroup$
2
$\begingroup$

Assuming what you have done up till the last part is correct, these hints I think will help you go further:

(1) Multiplying and dividing by $\sqrt{1+t}$

(2) Knowing the derivative of $\sinh^{-1} (\sqrt{t-1}) $ and $\sqrt{t+1}\sqrt{t-1}$

(3) Using partial integration

$\endgroup$

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.