What is the integral from-1/2 to 1/2 of 2/(1+4x^2) ?

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

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integral from-1/2 to 1/2 of 2/(1+4x^2)

\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{2}{1 + 4 x ^{2}} d x

\frac{π}{2}

Decimal Notation

1.57079 \dots

Solution steps

\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{2}{1 + 4 x ^{2}} d x

Take the constant out:

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

= 2 \cdot \int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{1}{1 + 4 x ^{2}} d x

Apply Integral Substitution

= 2 \cdot \int_{- 1}^{1} \frac{1}{2 ( 1 + u ^{2} )} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int_{- 1}^{1} \frac{1}{1 + u ^{2}} d u

Use the common integral:

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

= 2 \cdot \frac{1}{2} [arctan (u)]_{- 1}^{1}

Simplify

2 \cdot \frac{1}{2} [arctan (u)]_{- 1}^{1} : [arctan (u)]_{- 1}^{1}

= [arctan (u)]_{- 1}^{1}

Compute the boundaries:

\frac{π}{2}

= \frac{π}{2}

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