What is the integral of 1/((x+3/2)sqrt(x^2+3x+45/4)) ?

The integral of 1/((x+3/2)sqrt(x^2+3x+45/4)) is 1/3 ln|tan((arctan(1/3 x+1/2))/2)|+C

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integral of 1/((x+3/2)sqrt(x^2+3x+45/4))

\int \frac{1}{( x + \frac{3}{2} ) x ^{2} + 3 x + \frac{45}{4}} d x

\frac{1}{3} ln tan (\frac{arctan ( \frac{1}{3} x + \frac{1}{2} )}{2}) + C

Solution steps

\int \frac{1}{( x + \frac{3}{2} ) x ^{2} + 3 x + \frac{45}{4}} d x

Apply u-substitution

: \int \frac{1}{u u ^{2} + 9} d u

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

Apply Trigonometric Substitution

: \int \frac{sec ( v )}{3 tan ( v )} d v

= \int \frac{sec ( v )}{3 tan ( v )} d v

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{sec ( v )}{tan ( v )} d v

Apply u-substitution

: \int \frac{1}{w} d w

= \frac{1}{3} \cdot \int \frac{1}{w} d w

Use the common integral:

\int \frac{1}{w} d w = ln (∣ w ∣)

= \frac{1}{3} ln ∣ w ∣

Substitute back

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

Simplify

\frac{1}{3} ln tan (\frac{arctan ( \frac{1}{3} ( x + \frac{3}{2} ) )}{2}) : \frac{1}{3} ln tan (\frac{arctan ( \frac{1}{3} x + \frac{1}{2} )}{2})

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

Add a constant to the solution

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

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What is the integral of 1/((x+3/2)sqrt(x^2+3x+45/4)) ?
The integral of 1/((x+3/2)sqrt(x^2+3x+45/4)) is 1/3 ln|tan((arctan(1/3 x+1/2))/2)|+C