What is the integral from 0 to a of 1/(x^2+a^2) ?

The integral from 0 to a of 1/(x^2+a^2) is pi/(4a)

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integral from 0 to a of 1/(x^2+a^2)

\int_{0}^{a} \frac{1}{x ^{2} + a ^{2}} d x

\frac{π}{4 a}

Solution steps

\int_{0}^{a} \frac{1}{x ^{2} + a ^{2}} d x

Apply Integral Substitution

= \int_{0}^{1} \frac{1}{a ( 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}{a} \cdot \int_{0}^{1} \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= \frac{1}{a} [arctan (u)]_{0}^{1}

Compute the boundaries:

\frac{π}{4}

= \frac{1}{a} \cdot \frac{π}{4}

Simplify

= \frac{π}{4 a}

\frac{\partial}{\partial x} (1 - \frac{x}{y})

\int_{1}^{2} x e^{2 x^{2} - 3} d x

\int_{0}^{π} 10 t sin (3 t) d t

f (x) = x^{4} + 9 x^{2} - x, (1, 9)

x \to \infty lim (\frac{x \cdot ( x + 1 ) ^{x + 1}}{( x + 1 ) ^{x + 2}})

What is the integral from 0 to a of 1/(x^2+a^2) ?
The integral from 0 to a of 1/(x^2+a^2) is pi/(4a)