What is the integral of 3/(x^2+14x+97) ?

The integral of 3/(x^2+14x+97) is (sqrt(3))/4 arctan(1/(4sqrt(3))(x+7))+C

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integral of 3/(x^2+14x+97)

\int \frac{3}{x ^{2} + 14 x + 97} d x

\frac{3}{4} arctan (\frac{1}{4 3} (x + 7)) + C

Solution steps

\int \frac{3}{x ^{2} + 14 x + 97} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{x ^{2} + 14 x + 97} d x

Complete the square

x^{2} + 14 x + 97 : (x + 7)^{2} + 48

= 3 \cdot \int \frac{1}{( x + 7 ) ^{2} + 48} d x

Apply u-substitution

= 3 \cdot \int \frac{1}{u ^{2} + 48} d u

Apply Integral Substitution

= 3 \cdot \int \frac{1}{4 3 ( v ^{2} + 1 )} d v

Take the constant out:

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

= 3 \cdot \frac{1}{4 3} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= 3 \cdot \frac{1}{4 3} arctan (v)

Substitute back

= 3 \cdot \frac{1}{4 3} arctan (\frac{x + 7}{4 3})

Simplify

3 \cdot \frac{1}{4 3} arctan (\frac{x + 7}{4 3}) : \frac{3}{4} arctan (\frac{1}{4 3} (x + 7))

= \frac{3}{4} arctan (\frac{1}{4 3} (x + 7))

Add a constant to the solution

= \frac{3}{4} arctan (\frac{1}{4 3} (x + 7)) + C

What is the integral of 3/(x^2+14x+97) ?
The integral of 3/(x^2+14x+97) is (sqrt(3))/4 arctan(1/(4sqrt(3))(x+7))+C