Determine the integral by making an appropriate substitution.
-$\int \frac { x + \frac { 3 } { 2 } } { x ^ { 2 } + 3 x }$ dx
A) $\left( x ^ { 2 } + 3 x \right) ^ { 2 }$ + C
B) $\left( x ^ { 2 } + 3 x \right) ^ { - 1 }$ + C
C) $\frac { 1 } { 2 }$ ln $\left| x ^ { 2 } + 3 x \right|$ + C
D) ln $\left| x ^ { 2 } + 3 x \right|$ + C
E) none of these

Question

Determine the integral by making an appropriate substitution. 
-$\int \frac { x + \frac { 3 } { 2 } } { x ^ { 2 } + 3 x }$ dx
A) $\left( x ^ { 2 } + 3 x \right) ^ { 2 }$ + C
B) $\left( x ^ { 2 } + 3 x \right) ^ { - 1 }$ + C
C) $\frac { 1 } { 2 }$ ln $\left| x ^ { 2 } + 3 x \right|$ + C
D) ln $\left| x ^ { 2 } + 3 x \right|$ + C
E) none of these

Quizplus · Accepted Answer

The substitution $ u = x^2 + 3x $ simplifies the integral, leading to $\frac{1}{2} \ln |u|$ + C.

Determine the Integral by Making an Appropriate Substitution ∫x+32x2+3x\int \frac { x + \frac { 3 } { 2 } } { x ^ { 2 } + 3 x }∫x2+3xx+23​​

Determine the Integral by Making an Appropriate Substitution $\int \frac { x + \frac { 3 } { 2 } } { x ^ { 2 } + 3 x }$