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

The integral of 5/(sin(x)) is 5ln|tan(x/2)|+C

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integral of 5/(sin(x))

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

5 ln tan (\frac{x}{2}) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{u} d u

= 5 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 ln ∣ u ∣

Substitute back

u = tan (\frac{x}{2})

= 5 ln tan (\frac{x}{2})

Add a constant to the solution

= 5 ln tan (\frac{x}{2}) + C

4 2 x^{2} + 9

f (x) = - 3 x^{5} cos (x)

\int \frac{x ^{3}}{( 1 + x ^{4} ) ^{\frac{1}{2}}} d x

f (x) = x^{\frac{5}{2}} e^{x}

\int sec^{2} (x) csc (x) d x

What is the integral of 5/(sin(x)) ?
The integral of 5/(sin(x)) is 5ln|tan(x/2)|+C