What is the integral of (7t)/(5t^2-6) ?

The integral of (7t)/(5t^2-6) is 7/10 ln|5t^2-6|+C

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integral of (7t)/(5t^2-6)

\int \frac{7 t}{5 t ^{2} - 6} d t

\frac{7}{10} ln 5 t^{2} - 6 + C

Solution steps

\int \frac{7 t}{5 t ^{2} - 6} d t

Take the constant out:

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

= 7 \cdot \int \frac{t}{5 t ^{2} - 6} d t

Apply u-substitution

= 7 \cdot \int \frac{1}{10 u} d u

Take the constant out:

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

= 7 \cdot \frac{1}{10} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 7 \cdot \frac{1}{10} ln ∣ u ∣

Substitute back

u = 5 t^{2} - 6

= 7 \cdot \frac{1}{10} ln 5 t^{2} - 6

Simplify

7 \cdot \frac{1}{10} ln 5 t^{2} - 6 : \frac{7}{10} ln 5 t^{2} - 6

= \frac{7}{10} ln 5 t^{2} - 6

Add a constant to the solution

= \frac{7}{10} ln 5 t^{2} - 6 + C

What is the integral of (7t)/(5t^2-6) ?
The integral of (7t)/(5t^2-6) is 7/10 ln|5t^2-6|+C