What is the integral of 4/(16x^2+1) ?

The integral of 4/(16x^2+1) is arctan(4x)+C

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integral of 4/(16x^2+1)

\int \frac{4}{16 x ^{2} + 1} d x

arctan (4 x) + C

Solution steps

\int \frac{4}{16 x ^{2} + 1} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{16 x ^{2} + 1} d x

Apply Integral Substitution

= 4 \cdot \int \frac{1}{4 ( u ^{2} + 1 )} d u

Take the constant out:

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

= 4 \cdot \frac{1}{4} \cdot \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= 4 \cdot \frac{1}{4} arctan (u)

Substitute back

u = 4 x

= 4 \cdot \frac{1}{4} arctan (4 x)

Simplify

4 \cdot \frac{1}{4} arctan (4 x) : arctan (4 x)

= arctan (4 x)

Add a constant to the solution

= arctan (4 x) + C

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What is the integral of 4/(16x^2+1) ?
The integral of 4/(16x^2+1) is arctan(4x)+C