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

1) Before integrating, let's calculate:

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

ⓘ sin(x)/cos(x) = tan(x):

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

\frac{\tan (x)}{{\cos (x)}^{2}}

\frac{\tan (x)}{{\cos (x)}^{2}}

2) Let's integrate:

\int \frac{\tan (x)}{{\cos (x)}^{2}} d x

For the math expression:

\frac{\tan (x)}{{\cos (x)}^{2}}

, since (a^2)' = 2*a'*a, we try differentiating

{\tan (x)}^{2}

3) Let's perform the differentiation:

\frac{d}{d x} ({\tan (x)}^{2})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

\tan (x)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({\tan (x)}^{2})

2 \cdot \frac{d}{d x} (\tan (x)) \cdot \tan (x)

2 \cdot \frac{d}{d x} (\tan (x)) \cdot \tan (x)

ⓘ The derivative of tan(x) is 1/cos(x)^2:

\frac{d}{d x} (\tan (x))

\frac{1}{{\cos (x)}^{2}}

2 \cdot \frac{1}{{\cos (x)}^{2}} \cdot \tan (x)

ⓘ Multiplying a fraction by a term multiplies its numerator:

2 \cdot \frac{1}{{\cos (x)}^{2}} \cdot \tan (x)

\frac{2 \cdot 1 \cdot \tan (x)}{{\cos (x)}^{2}}

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2}}

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2}}

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

ⓘ The derivative of

\frac{1}{2} \cdot {\tan (x)}^{2}

\frac{\tan (x)}{{\cos (x)}^{2}}

, so

\frac{1}{2} \cdot {\tan (x)}^{2}

is an antiderivative of

\frac{\tan (x)}{{\cos (x)}^{2}}

\int \frac{\tan (x)}{{\cos (x)}^{2}} d x

\frac{1}{2} \cdot {\tan (x)}^{2}

\frac{1}{2} \cdot {\tan (x)}^{2}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{1}{2} \cdot {\tan (x)}^{2}

\frac{1 \cdot {\tan (x)}^{2}}{2}

\frac{{\tan (x)}^{2}}{2}

\frac{{\tan (x)}^{2}}{2}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

\frac{{\tan (x)}^{2}}{2} + C