An interesting trigonometric integral

Once while solving problems from Romanian Mathematical Olympiads I encountered a mathematical gem. The problem I found was asking to calcualte the following integral \[I=\int\frac{\cos x}{a\sin x+b\cos x}\,dx.\]

What do I find especially beautiful about this problem? It rewards the desire to examine a bit more than is asked initially!

Solution. Let's try to evaluate one more integral $J=\int{\frac{\sin x}{a\sin x+b\cos x}}\,dx.$

Can you guess the next step?

Yes, we will take advantage of the fact that the sum of integrals is the integral of sum. In other words: $aJ+bI=a\int{\frac{\sin x}{a\sin x+b\cos x}}\,dx+b\int{\frac{\cos x}{a\sin x+b\cos x}}\,dx\\=\int{\frac{a\sin x+b\cos x}{a\sin x+b\cos x}}\,dx=x+C.$

On other hand we can easily calculate integrals of the form $\int{\frac{f^\prime(x)}{f(x)}}\,dx$ . The derivative of $(a\sin x+b\cos x)$ is equal to $(a\cos x-b\sin x)$ , which can also be expressed as a linear combination of our two integrals, i.e. $aI-bJ=\int{\frac{a\cos x-b\sin x}{a\sin x+b\cos x}}\,dx=\ln{|a\sin x+b\cos x|}+C.$

Now we simply have to solve a system of linear equations. We obtain $I=\frac{a\ln{|a\sin x+b\cos x|}+bx}{a^2+b^2}+C.$ and $J=\frac{ax-\ln{|a\sin x+b\cos x|}}{a^2+b^2}+C.$

Has anyone of you seen problems with the similar ideas? How often while solving a problem you investigate possible variations of the conditions?

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

Try solving this the same way: $\Large I = \int \frac {a \sin x + b \cos x }{c \sin x + d \cos x} \,dx$

Pi Han Goh - 7 years, 6 months ago

Lets split the integral $I$ into the sum of two integrals: $I=a\int\frac{\sin x}{c\sin x+d\cos x}\,dx+b\int\frac{\cos x}{c\sin x+d\cos x}\,dx=aJ+bK.$
Now we reduced our problem to the previous one, because integrals $J,K$ are exactly the same as the one mentioned in the post.

Nicolae Sapoval - 7 years, 6 months ago

Smiley_Face.gif

Invitation (click the problem) $\int_0^{\Large\frac{\pi}{4}}\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\, \exp\left[\frac{x^2-1}{x^2+1}\right]\, dx$ .

Anastasiya Romanova - 7 years ago

This looks very useful, thanks for sharing Nicolae. :)

Pranav Arora - 7 years, 6 months ago

Thanks for sharing !!

Purvam Modi - 7 years, 6 months ago

Thanks for sharing

Rajath Krishna R - 7 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$