Question

$f\left(x\right)=\int \frac{3sinx+4cosx}{4sinx-3cosx}dx$
If the value of $f(-\frac{\pi }{2})$ is In (a), find a. Neglect the constant of integration while integrating
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Answer 1

In numerator and denominator we have sine and cosine functions. Their coefficients are either 3 or 4. We know that sine and cosine functions are derivative or integral of each other. With these hints and looking at the integral, we can see that derivative of denominator is the function in the numerator
So, we will make the substitution,
4 sinx - 3 cosx = t
We get ,
dt = (4 cosx + 3 sinx ) dx = Numerator
$\Rightarrow \int \frac{3sinx+4cosx}{4sinx-3cosx}dx=\int \frac{dt}{t}=In\left(t\right)$
Replacing t with 4 sinx - 3 cosx, we get
f(x) = ln|(4 sinx - 3 cosx)|, constant of integration is given as zero.
$\Rightarrow f\left(\frac{\pi }{2}\right)=In\left(4\right)=In\left(a\right)\Rightarrow a=4$