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

The integral of 1/(xsqrt(x^2+a^2)) is -1/(2a)(ln|(sqrt(x^2+a^2))/a+1|-ln|(sqrt(x^2+a^2))/a-1|)+C

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integral of 1/(xsqrt(x^2+a^2))

\int \frac{1}{x x ^{2} + a ^{2}} d x

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

Solution steps

\int \frac{1}{x x ^{2} + a ^{2}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Factor

w^{2} - 1 : - (- w^{2} + 1)

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

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

Substitute back

= \frac{1}{2} \cdot 2 \cdot \frac{1}{a} - \frac{ln \frac{x ^{2} + a ^{2}}{a} + 1}{2} - \frac{ln \frac{x ^{2} + a ^{2}}{a} - 1}{2}

Simplify

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

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

Add a constant to the solution

= - \frac{1}{2 a} (ln \frac{x ^{2} + a ^{2}}{a} + 1 - ln \frac{x ^{2} + a ^{2}}{a} - 1) + C

y^{^{'}} = x + 9 y

\frac{d}{d x} (x^{3} cos (x) - x^{3} sin (x))

x \to 2 lim (\frac{x ^{2}}{2 x - 4})

2 t^{3} - 15 t^{2} + 24 t

\frac{\partial}{\partial x} (\frac{x y}{x - y})

What is the integral of 1/(xsqrt(x^2+a^2)) ?
The integral of 1/(xsqrt(x^2+a^2)) is -1/(2a)(ln|(sqrt(x^2+a^2))/a+1|-ln|(sqrt(x^2+a^2))/a-1|)+C