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

解答

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

解答

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

求解步骤

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

使用换元积分法

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

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

使用三角换元法

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

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

提出常数:

\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

使用换元积分法

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

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

使用积分常见定则:

\int \frac{1}{w} d w = ln (∣ w ∣)

= \frac{1}{3} ln ∣ w ∣

代回

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

化简

\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})

解答补常数

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

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