What is the integral of 1/(x^2+3/4) ?

The integral of 1/(x^2+3/4) is 2/(sqrt(3))arctan((2x)/(sqrt(3)))+C

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integral of 1/(x^2+3/4)

\int \frac{1}{x ^{2} + \frac{3}{4}} d x

\frac{2}{3} arctan (\frac{2 x}{3}) + C

Solution steps

\int \frac{1}{x ^{2} + \frac{3}{4}} d x

Apply Integral Substitution

= \int \frac{2}{3 ( u ^{2} + 1 )} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= \frac{2}{3} arctan (u)

Substitute back

u = \frac{x}{\frac{3}{2}}

= \frac{2}{3} arctan (\frac{x}{\frac{3}{2}})

Simplify

= \frac{2}{3} arctan (\frac{2 x}{3})

Add a constant to the solution

= \frac{2}{3} arctan (\frac{2 x}{3}) + C

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

f (x) = - x^{3}, (- 1, 1)

\int_{- 2}^{1} 2 x d x

\frac{d y}{d x} = 5 x y

\int \frac{- 46 x + 158}{( x + 2 ) ( x - 3 ) ^{2}} d x

What is the integral of 1/(x^2+3/4) ?
The integral of 1/(x^2+3/4) is 2/(sqrt(3))arctan((2x)/(sqrt(3)))+C