What is the integral of 1/(usqrt(a^2-u^2)) ?

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

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integral of 1/(usqrt(a^2-u^2))

\int \frac{1}{u a ^{2} - u ^{2}} d u

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

Solution steps

\int \frac{1}{u a ^{2} - u ^{2}} d u

Apply u-substitution

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

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

Take the constant out:

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

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

Factor

- v^{2} + a^{2} : - (v^{2} - a^{2})

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

Take the constant out:

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

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

Apply Integral Substitution

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

Factor

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial z} (cos (x yz))

\frac{d y}{d t} = 1.4 y

\int \frac{1}{- x ^{2} - 1} d x

y^{^{'}} = \frac{( x y ^{3} )}{25}

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