What is the integral of (csc(7θ))/(csc(7θ)-sin(7θ)) ?

The integral of (csc(7θ))/(csc(7θ)-sin(7θ)) is 1/7 tan(7θ)+C

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integral of (csc(7θ))/(csc(7θ)-sin(7θ))

\int \frac{csc ( 7 θ )}{csc ( 7 θ ) - sin ( 7 θ )} d θ

\frac{1}{7} tan (7 θ) + C

Solution steps

\int \frac{csc ( 7 θ )}{csc ( 7 θ ) - sin ( 7 θ )} d θ

Rewrite using trig identities

= \int sec^{2} (7 θ) d θ

Apply u-substitution

: \int sec^{2} (u) \frac{1}{7} d u

= \int sec^{2} (u) \frac{1}{7} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{7} \cdot \int sec^{2} (u) d u

Use the common integral:

\int sec^{2} (u) d u = tan (u)

= \frac{1}{7} tan (u)

Substitute back

u = 7 θ

= \frac{1}{7} tan (7 θ)

Add a constant to the solution

= \frac{1}{7} tan (7 θ) + C

x \to 3 lim (4 x - 5)

\frac{d}{d x} (tan^{2} (n x))

- \frac{4}{5} sin (x)

y = 2 x^{2} - 8, y = - 5 + 5 x

\frac{d}{d t} (2 e^{- t^{2}} t)

What is the integral of (csc(7θ))/(csc(7θ)-sin(7θ)) ?
The integral of (csc(7θ))/(csc(7θ)-sin(7θ)) is 1/7 tan(7θ)+C