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

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

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

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

\frac{1}{7} tan (7 x) + C

Solution steps

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

Rewrite using trig identities

= \int sec^{2} (7 x) d x

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 x

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

Add a constant to the solution

= \frac{1}{7} tan (7 x) + C

\int_{- 1}^{1} \frac{1}{x ( x + 4 )} d x

\frac{d}{d x} (8 x - x^{2})

f (x) = 3 x^{7}

\frac{\partial}{\partial x} (\frac{3 x - t}{4 s + 5 t})

\frac{d}{d t} (4 t e^{- t})

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