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(x)=3\,\left(x + \frac{1}{2}\right)^2 + 2

f(x)=(-3)\,\frac{6\,\left(x + \frac{1}{2}\right)}{3\,\left(x + \frac{1}{2}\right)^2 + 2}

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

x\in\mathbb{R}

Give exact answers

(a)1 mark

Determine the indefinite integral of the target integrand

\int f(x)\,dx

\int f(x)\,dx=-3\,\log_e\left(3\,\left(x + \frac{1}{2}\right)^2 + 2\right)+C

\int f(x)\,dx=-3\,\frac{6\,\left(x + \frac{1}{2}\right)}{3\,\left(x + \frac{1}{2}\right)^2 + 2}\,\log_e\left(3\,\left(x + \frac{1}{2}\right)^2 + 2\right)+C

\int f(x)\,dx=-3\,\frac{3\,\left(x + \frac{1}{2}\right)^2 + 2}{6\,\left(x + \frac{1}{2}\right)}+C

\int f(x)\,dx=-3\,\log_e\left(3\,\left(x + \frac{1}{2}\right)^2 + 2\right)^2+C

\int f(x)\,dx=(-3)\,\frac{6\,\left(x + \frac{1}{2}\right)}{3\,\left(x + \frac{1}{2}\right)^2 + 2}+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(x)\,dx=K\log_e(g(x))+C

Coefficient of the logarithmic term