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

The integral of sqrt(1+1/(3x)) 1/(x^2) is -2((3x+1)/(3x))^{3/2}+C

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

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

- 2 (\frac{3 x + 1}{3 x})^{\frac{3}{2}} + C

Solution steps

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

Apply u-substitution

: \int - 3 u d u

= \int - 3 u d u

Take the constant out:

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

= - 3 \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= - 3 \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + \frac{1}{3 x}

= - 3 \cdot \frac{2}{3} (1 + \frac{1}{3 x})^{\frac{3}{2}}

Simplify

- 3 \cdot \frac{2}{3} (1 + \frac{1}{3 x})^{\frac{3}{2}} : - 2 (\frac{3 x + 1}{3 x})^{\frac{3}{2}}

= - 2 (\frac{3 x + 1}{3 x})^{\frac{3}{2}}

Add a constant to the solution

= - 2 (\frac{3 x + 1}{3 x})^{\frac{3}{2}} + C

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

y^{^{'}} = y^{2} - 2

What is the integral of sqrt(1+1/(3x)) 1/(x^2) ?
The integral of sqrt(1+1/(3x)) 1/(x^2) is -2((3x+1)/(3x))^{3/2}+C