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)=t^{2} + 2\,t + 5

f(t)=(-1)\,\frac{2\,t + 2}{t^{2} + 2\,t + 5}

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{t^{2} + 2\,t + 5}{2\,t + 2}+C

\int f(t)\,dt=-\log_e\left(t^{2} + 2\,t + 5\right)^2+C

\int f(t)\,dt=(-1)\,\frac{2\,t + 2}{t^{2} + 2\,t + 5}+C

\int f(t)\,dt=-\log_e\left(t^{2} + 2\,t + 5\right)+C

\int f(t)\,dt=-\frac{2\,t + 2}{t^{2} + 2\,t + 5}\,\log_e\left(t^{2} + 2\,t + 5\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