What is the integral of (3t)/(t^2+1) ?

The integral of (3t)/(t^2+1) is 3/2 ln|t^2+1|+C

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integral of (3t)/(t^2+1)

\int \frac{3 t}{t ^{2} + 1} d t

\frac{3}{2} ln t^{2} + 1 + C

Solution steps

\int \frac{3 t}{t ^{2} + 1} d t

Take the constant out:

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

= 3 \cdot \int \frac{t}{t ^{2} + 1} d t

Apply u-substitution

: \int \frac{1}{2 u} d u

= 3 \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

= 3 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = t^{2} + 1

= 3 \cdot \frac{1}{2} ln t^{2} + 1

Simplify

3 \cdot \frac{1}{2} ln t^{2} + 1 : \frac{3}{2} ln t^{2} + 1

= \frac{3}{2} ln t^{2} + 1

Add a constant to the solution

= \frac{3}{2} ln t^{2} + 1 + C

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What is the integral of (3t)/(t^2+1) ?
The integral of (3t)/(t^2+1) is 3/2 ln|t^2+1|+C