What is the integral of 2/(e^xsqrt(11+e^{2x))} ?

The integral of 2/(e^xsqrt(11+e^{2x))} is -(2sqrt(11+e^{2x)})/(11e^x)+C

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integral of 2/(e^xsqrt(11+e^{2x))}

\int \frac{2}{e ^{x} 11 + e ^{2 x}} d x

- \frac{2 11 + e ^{2 x}}{11 e ^{x}} + C

Solution steps

\int \frac{2}{e ^{x} 11 + e ^{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{1}{e ^{x} 11 + e ^{2 x}} d x

Apply u-substitution

: \int \frac{1}{u ^{2} 11 + u ^{2}} d u

= 2 \cdot \int \frac{1}{u ^{2} 11 + u ^{2}} d u

Apply Trigonometric Substitution

: \int \frac{sec ( v )}{11 tan ^{2} ( v )} d v

= 2 \cdot \int \frac{sec ( v )}{11 tan ^{2} ( v )} d v

Take the constant out:

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

= 2 \cdot \frac{1}{11} \cdot \int \frac{sec ( v )}{tan ^{2} ( v )} d v

Rewrite using trig identities

= 2 \cdot \frac{1}{11} \cdot \int \frac{cos ( v )}{sin ^{2} ( v )} d v

Apply u-substitution

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

= 2 \cdot \frac{1}{11} \cdot \int \frac{1}{w ^{2}} d w

Apply the Power Rule

: - \frac{1}{w}

= 2 \cdot \frac{1}{11} (- \frac{1}{w})

Substitute back

= 2 \cdot \frac{1}{11} - \frac{1}{sin ( arctan ( \frac{1}{11} e ^{x} ) )}

Simplify

2 \cdot \frac{1}{11} - \frac{1}{sin ( arctan ( \frac{1}{11} e ^{x} ) )} : - \frac{2 11 + e ^{2 x}}{11 e ^{x}}

= - \frac{2 11 + e ^{2 x}}{11 e ^{x}}

Add a constant to the solution

= - \frac{2 11 + e ^{2 x}}{11 e ^{x}} + C

\int 7 cos^{3} (3 x) d x

x y^{^{'}} + 3 y = 0

x \to 0 + lim (3)

f (x) = cos (2 sin (x))

y = ln (e^{x} + x e^{x})

What is the integral of 2/(e^xsqrt(11+e^{2x))} ?
The integral of 2/(e^xsqrt(11+e^{2x))} is -(2sqrt(11+e^{2x)})/(11e^x)+C