1 of 8 0-day streak

Integration by recognition with logarithmic components

Question 1 of 8

No calculator

Use the exact function data shown below

g(t)=3\,t^{2} + 9\,t + \frac{31}{4}

f(t)=(1)\,\frac{6\,t + 9}{3\,t^{2} + 9\,t + \frac{31}{4}}

The inner expression is positive for every real value of the variable

t\in\mathbb{R}

Give exact answers

(a)1 mark

Determine the indefinite integral of the target integrand

\int f(t)\,dt

\int f(t)\,dt=\frac{6\,t + 9}{3\,t^{2} + 9\,t + \frac{31}{4}}\,\log_e\left(3\,t^{2} + 9\,t + \frac{31}{4}\right)+C

\int f(t)\,dt=(1)\,\frac{6\,t + 9}{3\,t^{2} + 9\,t + \frac{31}{4}}+C

\int f(t)\,dt=\log_e\left(3\,t^{2} + 9\,t + \frac{31}{4}\right)^2+C

\int f(t)\,dt=\frac{3\,t^{2} + 9\,t + \frac{31}{4}}{6\,t + 9}+C

\int f(t)\,dt=\log_e\left(3\,t^{2} + 9\,t + \frac{31}{4}\right)+C

(b)1 mark

Enter the constant K in the stated primitive form. Keep the supplied inner function g unchanged. Do not move factors into or out of the logarithm

\int f(t)\,dt=K\log_e(g(t))+C

Coefficient of the logarithmic term