What is the integral of (5+sec^2(x))/(5x+tan(x)) ?

The integral of (5+sec^2(x))/(5x+tan(x)) is ln|5x+tan(x)|+C

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integral of (5+sec^2(x))/(5x+tan(x))

\int \frac{5 + sec ^{2} ( x )}{5 x + tan ( x )} d x

ln ∣ 5 x + tan (x) ∣ + C

Solution steps

\int \frac{5 + sec ^{2} ( x )}{5 x + tan ( x )} d x

Apply u-substitution

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = 5 x + tan (x)

= ln ∣ 5 x + tan (x) ∣

Add a constant to the solution

= ln ∣ 5 x + tan (x) ∣ + C

What is the integral of (5+sec^2(x))/(5x+tan(x)) ?
The integral of (5+sec^2(x))/(5x+tan(x)) is ln|5x+tan(x)|+C