1 of 8 0-day streak

Anti-differentiate reciprocal and linear-reciprocal functions

Question 1 of 8

No calculator

The function to be anti-differentiated is given below

q(z)=\frac{-3}{\left(z\right)}

\int \frac{-3}{\left(z\right)}\,dz

The stated branch of the linear denominator is shown below

z < 0

(a)1 mark

Find the full family of functions with this derivative

The arbitrary additive constant is included in the family

F(z)=-3\,\log_e\left(z\right) + C

F(z)=-3\,\log_e\left|z\right|

F(z)=-3\,\left(z\right)^{0} + C

F(z)=-3\,\log_e\left|z\right| + C

(b)3 marks

Write the logarithmic term in the displayed form. Keep the linear expression inside the absolute value exactly equal to the original denominator, without rescaling it or reversing its sign. Enter K, A and B

K\log_e\left|A\,z+B\right|

Logarithmic coefficient K Coefficient A of z in the original denominator Constant B in the original denominator