What is the integral of (t+1)/(7t^2+14t-5) ?

The integral of (t+1)/(7t^2+14t-5) is 1/14 ln|7t^2+14t-5|+C

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integral of (t+1)/(7t^2+14t-5)

\int \frac{t + 1}{7 t ^{2} + 14 t - 5} d t

\frac{1}{14} ln 7 t^{2} + 14 t - 5 + C

Solution steps

\int \frac{t + 1}{7 t ^{2} + 14 t - 5} d t

Apply u-substitution

: \int \frac{1}{14 u} d u

= \int \frac{1}{14 u} d u

Take the constant out:

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

= \frac{1}{14} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{14} ln ∣ u ∣

Substitute back

u = 7 t^{2} + 14 t - 5

= \frac{1}{14} ln 7 t^{2} + 14 t - 5

Add a constant to the solution

= \frac{1}{14} ln 7 t^{2} + 14 t - 5 + C

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What is the integral of (t+1)/(7t^2+14t-5) ?
The integral of (t+1)/(7t^2+14t-5) is 1/14 ln|7t^2+14t-5|+C