What is the integral of 1/(3+sqrt(2x)) ?

The integral of 1/(3+sqrt(2x)) is 3+sqrt(2)sqrt(x)-3ln|3+sqrt(2)sqrt(x)|+C

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integral of 1/(3+sqrt(2x))

\int \frac{1}{3 + 2 x} d x

3 + 2 x - 3 ln 3 + 2 x + C

Solution steps

\int \frac{1}{3 + 2 x} d x

Apply u-substitution

: \int \frac{u - 3}{u} d u

= \int \frac{u - 3}{u} d u

Expand

\frac{u - 3}{u} : 1 - \frac{3}{u}

= \int 1 - \frac{3}{u} 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

= \int 1 d u - \int \frac{3}{u} d u

\int 1 d u = u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= u - 3 ln ∣ u ∣

Substitute back

u = 3 + 2 x

= 3 + 2 x - 3 ln 3 + 2 x

Simplify

3 + 2 x - 3 ln 3 + 2 x : 3 + 2 x - 3 ln 3 + 2 x

= 3 + 2 x - 3 ln 3 + 2 x

Add a constant to the solution

= 3 + 2 x - 3 ln 3 + 2 x + C

\int - 1 sin^{3} (x) cos^{2} (x) d x

x \to \infty lim (x)

\int x (tan (x))^{2} d x

x^{2} + 1, 3 - x, x = 0

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

What is the integral of 1/(3+sqrt(2x)) ?
The integral of 1/(3+sqrt(2x)) is 3+sqrt(2)sqrt(x)-3ln|3+sqrt(2)sqrt(x)|+C