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 Complex looking simple integral (Posted on 2023-05-26)
Evaluate this integral.
```   3sin(x)+4
∫ ------------- dx
(3+4sin(x))2
```

 See The Solution Submitted by K Sengupta Rating: 5.0000 (1 votes)

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 Solution Comment 1 of 1
The trick to this integral is to start by multiplying the numerator and denominator by sec^2(x). Then we get
Integ (3*tan(x)*sec(x)+4*sec^2(x))/(3*sec(x)+4*tan(x))^2 dx

Now utilize the substitution u = 3*sec(x)+4*tan(x); du = 3*tan(x)*sec(x)+4*sec^2(x) dx.  Then:
Integ 1/u^2 du -> -1/u + C

Now just back-substitute to get
-1/(3*sec(x)+4*tan(x)) + C

One little last thing is to multiply the numerator and denominator by cos(x), to put the result back into terms of sin(x).  Then we have our final answer:
-cos(x)/(3+4*sin(x)) + C

 Posted by Brian Smith on 2023-05-26 23:00:39

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