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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Apply u-substitution

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

Simplify

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

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

Add a constant to the solution

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

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