What is the integral of 5/(sqrt(x^2+4x+13)) ?

The integral of 5/(sqrt(x^2+4x+13)) is 5ln(1/3 |x+2+sqrt(x^2+4x+13)|)+C

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integral of 5/(sqrt(x^2+4x+13))

\int \frac{5}{x ^{2} + 4 x + 13} d x

5 ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13) + C

Solution steps

\int \frac{5}{x ^{2} + 4 x + 13} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{x ^{2} + 4 x + 13} d x

Complete the square

x^{2} + 4 x + 13 : (x + 2)^{2} + 9

= 5 \cdot \int \frac{1}{( x + 2 ) ^{2} + 9} d x

Apply u-substitution

= 5 \cdot \int \frac{1}{u ^{2} + 9} d u

Apply Trigonometric Substitution

= 5 \cdot \int sec (v) d v

Use the common integral:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= 5 ln ∣ tan (v) + sec (v) ∣

Substitute back

= 5 ln tan (arctan (\frac{1}{3} (x + 2))) + sec (arctan (\frac{1}{3} (x + 2)))

Simplify

5 ln tan (arctan (\frac{1}{3} (x + 2))) + sec (arctan (\frac{1}{3} (x + 2))) : 5 ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13)

= 5 ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13)

Add a constant to the solution

= 5 ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13) + C

What is the integral of 5/(sqrt(x^2+4x+13)) ?
The integral of 5/(sqrt(x^2+4x+13)) is 5ln(1/3 |x+2+sqrt(x^2+4x+13)|)+C