What is the integral of (5sin^3(x))/(cos(x)) ?

The integral of (5sin^3(x))/(cos(x)) is 5(-ln|cos(x)|+(cos^2(x))/2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (5sin^3(x))/(cos(x))

\int \frac{5 sin ^{3} ( x )}{cos ( x )} d x

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

Solution steps

\int \frac{5 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

= 5 \cdot \int \frac{sin ^{3} ( x )}{cos ( x )} d x

Rewrite using trig identities

= 5 \cdot \int \frac{( 1 - cos ^{2} ( x ) ) sin ( x )}{cos ( x )} d x

Apply u-substitution

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

= 5 \cdot \int - \frac{1 - u ^{2}}{u} d u

Expand

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

= 5 \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

= 5 (- \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}

= 5 (- ln ∣ u ∣ + \frac{u ^{2}}{2})

Substitute back

u = cos (x)

= 5 (- ln ∣ cos (x) ∣ + \frac{cos ^{2} ( x )}{2})

Add a constant to the solution

= 5 (- ln ∣ cos (x) ∣ + \frac{cos ^{2} ( x )}{2}) + C

\frac{\partial}{\partial x} (x^{3} y)

\int (x^{8} + 1) e^{x^{9} + 9 x} d x

y^{^{''}} + 19 y = 0

x \to - 5 - lim (\frac{∣ 5 - x ∣}{5 - x})

\int \frac{x ^{2} + 36 x - 9}{x ^{3} - 9 x} d x

What is the integral of (5sin^3(x))/(cos(x)) ?
The integral of (5sin^3(x))/(cos(x)) is 5(-ln|cos(x)|+(cos^2(x))/2)+C