What is the integral of (11)/(5+11^{x2)} ?

The integral of (11)/(5+11^{x2)} is (11(-ln|121^x+5|+2ln(11)x))/(10ln(11))+C

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integral of (11)/(5+11^{x2)}

\int \frac{11}{5 + 1 1 ^{x 2}} d x

\frac{11 ( - ln ∣ 12 1 ^{x} + 5 ∣ + 2 ln ( 11 ) x )}{10 ln ( 11 )} + C

Solution steps

\int \frac{11}{5 + 1 1 ^{x \cdot 2}} d x

Take the constant out:

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

= 11 \cdot \int \frac{1}{5 + 1 1 ^{x \cdot 2}} d x

Apply u-substitution

: \int \frac{1}{2 ln ( 11 ) \cdot 12 1 ^{\frac{l n ( u - 5 )}{2 l n ( 11 )}} u} d u

= 11 \cdot \int \frac{1}{2 ln ( 11 ) \cdot 12 1 ^{\frac{l n ( u - 5 )}{2 l n ( 11 )}} u} d u

Take the constant out:

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

= 11 \cdot \frac{1}{2 ln ( 11 )} \cdot \int \frac{1}{12 1 ^{\frac{l n ( u - 5 )}{2 l n ( 11 )}} u} d u

Apply u-substitution

: \int \frac{1}{( v + 5 ) v} d v

= 11 \cdot \frac{1}{2 ln ( 11 )} \cdot \int \frac{1}{( v + 5 ) v} d v

Take the partial fraction of

\frac{1}{( v + 5 ) v} : - \frac{1}{5 ( v + 5 )} + \frac{1}{5 v}

= 11 \cdot \frac{1}{2 ln ( 11 )} \cdot \int - \frac{1}{5 ( v + 5 )} + \frac{1}{5 v} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 11 \cdot \frac{1}{2 ln ( 11 )} (- \int \frac{1}{5 ( v + 5 )} d v + \int \frac{1}{5 v} d v)

\int \frac{1}{5 ( v + 5 )} d v = \frac{1}{5} ln ∣ v + 5 ∣

\int \frac{1}{5 v} d v = \frac{1}{5} ln ∣ v ∣

= 11 \cdot \frac{1}{2 ln ( 11 )} (- \frac{1}{5} ln ∣ v + 5 ∣ + \frac{1}{5} ln ∣ v ∣)

Substitute back

= 11 \cdot \frac{1}{2 ln ( 11 )} (- \frac{1}{5} ln 12 1^{\frac{l n ( 5 + 1 1 ^{x \cdot 2} - 5 )}{2 l n ( 11 )}} + 5 + \frac{1}{5} ln 12 1^{\frac{l n ( 5 + 1 1 ^{x \cdot 2} - 5 )}{2 l n ( 11 )}})

Simplify

11 \cdot \frac{1}{2 ln ( 11 )} (- \frac{1}{5} ln 12 1^{\frac{l n ( 5 + 1 1 ^{x \cdot 2} - 5 )}{2 l n ( 11 )}} + 5 + \frac{1}{5} ln 12 1^{\frac{l n ( 5 + 1 1 ^{x \cdot 2} - 5 )}{2 l n ( 11 )}}) : \frac{11 ( - ln ∣ 12 1 ^{x} + 5 ∣ + 2 ln ( 11 ) x )}{10 ln ( 11 )}

= \frac{11 ( - ln ∣ 12 1 ^{x} + 5 ∣ + 2 ln ( 11 ) x )}{10 ln ( 11 )}

Add a constant to the solution

= \frac{11 ( - ln ∣ 12 1 ^{x} + 5 ∣ + 2 ln ( 11 ) x )}{10 ln ( 11 )} + C

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What is the integral of (11)/(5+11^{x2)} ?
The integral of (11)/(5+11^{x2)} is (11(-ln|121^x+5|+2ln(11)x))/(10ln(11))+C