What is the integral of (sqrt(1+x))/(2x) ?

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

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integral of (sqrt(1+x))/(2x)

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

- \frac{1}{2} ln 1 + x + 1 + \frac{1}{2} ln 1 + x - 1 + 1 + x + C

Solution steps

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

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 + x}{x} d x

Apply u-substitution

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

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

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{u ^{2}}{u ^{2} - 1} d u

Factor

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

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

Take the constant out:

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

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

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

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

Apply the Sum Rule:

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

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

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

\int 1 d u = u

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

Substitute back

u = 1 + x

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

Simplify

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

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

Add a constant to the solution

= - \frac{1}{2} ln 1 + x + 1 + \frac{1}{2} ln 1 + x - 1 + 1 + x + C

y = x, y = 12 - 4 x, x = 0, x = 3

y^{^{'}} = \frac{4 x ^{3} + 1}{2 y - 6}, y (1) = 2

\frac{d}{d x} (a^{2} x^{2})

\int 3 cos (5 x) d x

x \to \infty lim (an)

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