What is the integral of (2t)/(t^2+4) ?

The integral of (2t)/(t^2+4) is ln|t^2+4|+C

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integral of (2t)/(t^2+4)

\int \frac{2 t}{t ^{2} + 4} d t

ln t^{2} + 4 + C

Solution steps

\int \frac{2 t}{t ^{2} + 4} d t

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int \frac{1}{2 u} 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 \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = t^{2} + 4

= 2 \cdot \frac{1}{2} ln t^{2} + 4

Simplify

2 \cdot \frac{1}{2} ln t^{2} + 4 : ln t^{2} + 4

= ln t^{2} + 4

Add a constant to the solution

= ln t^{2} + 4 + C

What is the integral of (2t)/(t^2+4) ?
The integral of (2t)/(t^2+4) is ln|t^2+4|+C