integral of-sin(x)sec(x)tan(x) — step-by-step solution

The secant is read as

\sec (u) = \frac{1}{\cos (u)}

, on each interval where

\cos (u) \neq 0

⚠ -sin(x)(1/cos(x))tan(x) is read as -sin(x)*(1/cos(x))*tan(x): use (...)*(...) for multiplication, use (...)^(...) for a power

Before integrating, let's calculate:

- \sin (x) \cdot \frac{1}{\cos (x)} \cdot \tan (x)

ℹ

- \sin (x) \cdot \frac{1}{\cos (x)} \cdot \tan (x)

\frac{- \sin (x) \cdot \tan (x)}{\cos (x)}

‖

\frac{- \sin (x) \cdot \tan (x)}{\cos (x)}

- \tan (x) \cdot \tan (x)

- \tan (x) \cdot \tan (x)

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

Let's compute the integral:

\int(- {\tan (x)}^{2})dx

We reduce the square using the identity

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

- \int(\frac{1}{{\cos (x)}^{2}} - 1)dx

For the math expression:

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

, we know the derivative formula:

\frac{d}{d x} (\tan (a)) = \frac{1}{{\cos (a)}^{2}}

, so we try differentiating with tan

Let's differentiate:

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

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

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

- (\tan (x) + \int(-1)dx)

- \tan (x) - \int(-1)dx

- \tan (x) - (-1 \cdot x)

- \tan (x) + x

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

x - \tan (x) + C