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$\int\frac{dx}{\cos^3x-\sin^3x}$


Let $I=\int\frac{dx}{\cos^3x-\sin^3x}=\int\frac{dx}{(\cos x-\sin x)(\cos^2 x+\sin^2 x+\sin x\cos x)}$

But it does not seem to be solved further by this method,so i tried another method.

$I=\int\frac{dx}{\cos^3x-\sin^3x}=\int\frac{\csc^3 xdx}{\cot^3x-1}=\int\frac{\csc^2 x \csc xdx}{\cot^3x-1}=\int\frac{\csc^2 x \sqrt{1+\cot^2x}dx}{\cot^3x-1}$

Put $\cot x=t\implies -\csc^2 x dx=dt$

$I=\int\frac{-\sqrt{1+t^2}dt}{t^3-1}=\int\frac{-\sqrt{1+t^2}dt}{(t-1)(t^2+t+1)}$

But i am stuck here.

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$$\cos^3x-\sin^3x=(\cos x-\sin x)(1+\sin x\cos x)=\dfrac{(\cos x-\sin x)\{3-(\cos x-\sin x)^2\}}2$$

Writing $\cos x-\sin x=t$ and using Partial Fraction,

$$\dfrac1{t(3-t^2)}=\dfrac1{3t}+\dfrac t{3(3-t^2)}$$

$$\implies\dfrac3{\cos^3x-\sin^3x}=\dfrac1{\cos x-\sin x}+\dfrac{\cos x-\sin x}{3-(\cos x-\sin x)^2}$$

The first integral can be managed easily

For the second $$\text{as }\int(\cos x-\sin x)dx=\sin x+\cos x$$ and as $$(\cos x-\sin x)^2+(\sin x+\cos x)^2=2$$

write $$3-(\cos x-\sin x)^2=1+(\sin x+\cos x)^2$$ and replace $\sin x+\cos x$ with $u$

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