What is the integral of (3sin^3(x))/(cos(x)) ?

The integral of (3sin^3(x))/(cos(x)) is 3(-ln|cos(x)|+(cos^2(x))/2)+C

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integral of (3sin^3(x))/(cos(x))

\int \frac{3 sin ^{3} ( x )}{cos ( x )} d x

3 (- ln ∣ cos (x) ∣ + \frac{cos ^{2} ( x )}{2}) + C

Solution steps

\int \frac{3 sin ^{3} ( x )}{cos ( x )} d x

Take the constant out:

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

= 3 \cdot \int \frac{sin ^{3} ( x )}{cos ( x )} d x

Rewrite using trig identities

= 3 \cdot \int \frac{( 1 - cos ^{2} ( x ) ) sin ( x )}{cos ( x )} d x

Apply u-substitution

: \int - \frac{1 - u ^{2}}{u} d u

= 3 \cdot \int - \frac{1 - u ^{2}}{u} d u

Expand

- \frac{1 - u ^{2}}{u} : - \frac{1}{u} + u

= 3 \cdot \int - \frac{1}{u} + u d u

Apply the Sum Rule:

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

= 3 (- \int \frac{1}{u} d u + \int u d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int u d u = \frac{u ^{2}}{2}

= 3 (- ln ∣ u ∣ + \frac{u ^{2}}{2})

Substitute back

u = cos (x)

= 3 (- ln ∣ cos (x) ∣ + \frac{cos ^{2} ( x )}{2})

Add a constant to the solution

= 3 (- ln ∣ cos (x) ∣ + \frac{cos ^{2} ( x )}{2}) + C

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\frac{d}{d x} (sin (ln (3 x^{2} + x)))

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\int \frac{1}{x \cdot 1 - x ^{2}} d x

What is the integral of (3sin^3(x))/(cos(x)) ?
The integral of (3sin^3(x))/(cos(x)) is 3(-ln|cos(x)|+(cos^2(x))/2)+C