What is the integral of tan(2x)sec^2(x) ?

The integral of tan(2x)sec^2(x) is -ln|1-tan^2(x)|+C

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integral of tan(2x)sec^2(x)

\int tan (2 x) sec^{2} (x) d x

- ln 1 - tan^{2} (x) + C

Solution steps

\int tan (2 x) sec^{2} (x) d x

Rewrite using trig identities

= \int \frac{2 sin ( x )}{cos ( x ) cos ( 2 x )} d x

Take the constant out:

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

= 2 \cdot \int \frac{sin ( x )}{cos ( x ) cos ( 2 x )} d x

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

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

Apply u-substitution

: \int \frac{tan ( \frac{u}{2} )}{2 cos ( u )} d u

= 2 \cdot \int \frac{tan ( \frac{u}{2} )}{2 cos ( u )} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{tan ( \frac{u}{2} )}{cos ( u )} d u

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{1}{2 w} d w

= 2 \cdot \frac{1}{2} \cdot 2 \cdot \int - \frac{1}{2 w} d w

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot 2 (- \frac{1}{2} \cdot \int \frac{1}{w} d w)

Use the common integral:

\int \frac{1}{w} d w = ln (∣ w ∣)

= 2 \cdot \frac{1}{2} \cdot 2 (- \frac{1}{2} ln ∣ w ∣)

Substitute back

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

Simplify

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

= - ln 1 - tan^{2} (x)

Add a constant to the solution

= - ln 1 - tan^{2} (x) + C

\frac{\partial}{\partial x} (x^{0.25} y^{0.25})

\int (\frac{4 x ^{19} - 5 x ^{- 8}}{x ^{2}}) d x

\frac{d y}{d x} = (\frac{y ^{3}}{x ^{2}})

\int x (x^{2} + 2)^{6} d x

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What is the integral of tan(2x)sec^2(x) ?
The integral of tan(2x)sec^2(x) is -ln|1-tan^2(x)|+C