What is the integral of 1/(4x^2+7) ?

The integral of 1/(4x^2+7) is 1/(2sqrt(7))arctan(2/(sqrt(7))x)+C

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integral of 1/(4x^2+7)

\int \frac{1}{4 x ^{2} + 7} d x

\frac{1}{2 7} arctan (\frac{2}{7} x) + C

Solution steps

\int \frac{1}{4 x ^{2} + 7} d x

Apply Integral Substitution

= \int \frac{1}{2 7 ( 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 7} \cdot \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= \frac{1}{2 7} arctan (u)

Substitute back

u = \frac{2}{7} x

= \frac{1}{2 7} arctan (\frac{2}{7} x)

Add a constant to the solution

= \frac{1}{2 7} arctan (\frac{2}{7} x) + C

(x, y) \to (0, 0) lim (\frac{2 x}{x ^{2} + x + y ^{2}})

\int \frac{3 x + 5}{x ^{3} - x ^{2} - x + 1} d x

\frac{d}{d x} (arctan (sin (x^{2} + \frac{π}{x})))

y^{^{''}} + y^{^{'}} + y = t

\frac{d}{d x} (6 x^{2} - 4 x)

What is the integral of 1/(4x^2+7) ?
The integral of 1/(4x^2+7) is 1/(2sqrt(7))arctan(2/(sqrt(7))x)+C