What is the integral of (10t)/(t^2+9) ?

The integral of (10t)/(t^2+9) is 5ln|t^2+9|+C

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integral of (10t)/(t^2+9)

\int \frac{10 t}{t ^{2} + 9} d t

5 ln t^{2} + 9 + C

Solution steps

\int \frac{10 t}{t ^{2} + 9} d t

Take the constant out:

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

= 10 \cdot \int \frac{t}{t ^{2} + 9} d t

Apply u-substitution

= 10 \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

= 10 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 10 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = t^{2} + 9

= 10 \cdot \frac{1}{2} ln t^{2} + 9

Simplify

10 \cdot \frac{1}{2} ln t^{2} + 9 : 5 ln t^{2} + 9

= 5 ln t^{2} + 9

Add a constant to the solution

= 5 ln t^{2} + 9 + C

What is the integral of (10t)/(t^2+9) ?
The integral of (10t)/(t^2+9) is 5ln|t^2+9|+C