What is the integral of 1/(1-sin(5x)) ?

The integral of 1/(1-sin(5x)) is 1/5 tan(5x)+1/5 sec(5x)+C

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integral of 1/(1-sin(5x))

\int \frac{1}{1 - sin ( 5 x )} d x

\frac{1}{5} tan (5 x) + \frac{1}{5} sec (5 x) + C

Solution steps

\int \frac{1}{1 - sin ( 5 x )} d x

Multiply by the conjugate of

1 - sin (5 x) : \frac{cos ^{2} ( 5 x )}{1 + sin ( 5 x )}

= \int \frac{1}{\frac{c o s ^{2} ( 5 x )}{1 + s i n ( 5 x )}} d x

Expand

\frac{1}{\frac{c o s ^{2} ( 5 x )}{1 + s i n ( 5 x )}} : \frac{1}{cos ^{2} ( 5 x )} + \frac{sin ( 5 x )}{cos ^{2} ( 5 x )}

= \int \frac{1}{cos ^{2} ( 5 x )} + \frac{sin ( 5 x )}{cos ^{2} ( 5 x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{cos ^{2} ( 5 x )} d x + \int \frac{sin ( 5 x )}{cos ^{2} ( 5 x )} d x

\int \frac{1}{cos ^{2} ( 5 x )} d x = \frac{1}{5} tan (5 x)

\int \frac{sin ( 5 x )}{cos ^{2} ( 5 x )} d x = \frac{1}{5} sec (5 x)

= \frac{1}{5} tan (5 x) + \frac{1}{5} sec (5 x)

Add a constant to the solution

= \frac{1}{5} tan (5 x) + \frac{1}{5} sec (5 x) + C

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What is the integral of 1/(1-sin(5x)) ?
The integral of 1/(1-sin(5x)) is 1/5 tan(5x)+1/5 sec(5x)+C