What is the integral of (2tan(x))/((1+2tan^2(x))) ?

The integral of (2tan(x))/((1+2tan^2(x))) is ln|1+sin^2(x)|+C

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integral of (2tan(x))/((1+2tan^2(x)))

integral

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

ln 1 + sin^{2} (x) + C

Solution steps

\int \frac{2 tan ( x )}{1 + 2 tan ^{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{tan ( x )}{1 + 2 tan ^{2} ( x )} d x

Rewrite using trig identities

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

Rewrite using trig identities

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

Rewrite using trig identities

= 2 \cdot \int \frac{sin ( 2 x )}{2 ( 1 + sin ^{2} ( x ) )} d x

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{sin ( 2 x )}{1 + sin ^{2} ( x )} d x

Apply u-substitution

: \int \frac{1}{u} d u

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 1 + sin^{2} (x)

= 2 \cdot \frac{1}{2} ln 1 + sin^{2} (x)

Simplify

2 \cdot \frac{1}{2} ln 1 + sin^{2} (x) : ln 1 + sin^{2} (x)

= ln 1 + sin^{2} (x)

Add a constant to the solution

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

\frac{( x + 1 ) ^{2}}{( x - 1 ) ^{2}}

tan (2 - 3 x)

e^{x^{2}} x^{2}

3 x^{2} - 9 x + 5

\frac{e ^{x}}{3}

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