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)=\frac{1}{2}\,\left(x - \frac{3}{2}\right)^2 + 4

f(x)=(2)\,\frac{\left(x - \frac{3}{2}\right)}{\frac{1}{2}\,\left(x - \frac{3}{2}\right)^2 + 4}

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=2\,\log_e\left(\frac{1}{2}\,\left(x - \frac{3}{2}\right)^2 + 4\right)^2+C

\int f(x)\,dx=2\,\log_e\left(\frac{1}{2}\,\left(x - \frac{3}{2}\right)^2 + 4\right)+C

\int f(x)\,dx=(2)\,\frac{\left(x - \frac{3}{2}\right)}{\frac{1}{2}\,\left(x - \frac{3}{2}\right)^2 + 4}+C

\int f(x)\,dx=2\,\frac{\left(x - \frac{3}{2}\right)}{\frac{1}{2}\,\left(x - \frac{3}{2}\right)^2 + 4}\,\log_e\left(\frac{1}{2}\,\left(x - \frac{3}{2}\right)^2 + 4\right)+C

\int f(x)\,dx=2\,\frac{\frac{1}{2}\,\left(x - \frac{3}{2}\right)^2 + 4}{\left(x - \frac{3}{2}\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(x)\,dx=K\log_e(g(x))+C

Coefficient of the logarithmic term