What is the integral of (7csc^3(x))/(tan(x)) ?

The integral of (7csc^3(x))/(tan(x)) is -7/3 csc^3(x)+C

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integral of (7csc^3(x))/(tan(x))

\int \frac{7 csc ^{3} ( x )}{tan ( x )} d x

- \frac{7}{3} csc^{3} (x) + C

Solution steps

\int \frac{7 csc ^{3} ( x )}{tan ( x )} d x

Take the constant out:

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

= 7 \cdot \int \frac{csc ^{3} ( x )}{tan ( x )} d x

Use the basic trigonometric identity:

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

= 7 \cdot \int csc^{3} (x) cot (x) d x

Apply u-substitution

: \int - u^{2} d u

= 7 \cdot \int - u^{2} d u

Take the constant out:

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

= 7 (- \int u^{2} d u)

Apply the Power Rule

: \frac{u ^{3}}{3}

= 7 (- \frac{u ^{3}}{3})

Substitute back

u = csc (x)

= 7 (- \frac{csc ^{3} ( x )}{3})

Simplify

7 (- \frac{csc ^{3} ( x )}{3}) : - \frac{7}{3} csc^{3} (x)

= - \frac{7}{3} csc^{3} (x)

Add a constant to the solution

= - \frac{7}{3} csc^{3} (x) + C

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What is the integral of (7csc^3(x))/(tan(x)) ?
The integral of (7csc^3(x))/(tan(x)) is -7/3 csc^3(x)+C