Integrating Special Functions (DP IB Applications & Interpretation (AI): HL): Revision Note

The three standard integrals above are all in the exam formula booklet.

Make sure you have a copy of the formula booklet during revision but don't try to memorise everything in the formula booklet.

Instead, make sure you are familiar with the layout of the formula booklet. That way you’ll be able to quickly locate whatever you are after.

For formulae you think you have remembered, use the booklet to double check.

a)  Find, in the form$\mathrm{F}\left(x\right)+c$, an expression for each integral

1. $\int \cos x\mathrm{d}x$

2. $\int \frac{1}{{\cos }^2\left(3x-{\displaystyle \frac{\pi }{3}}\right)}\mathrm{d}x$

Answer:

b)  A curve has equation$y=\int 2\sin \left(2x+\frac{\pi }{6}\right)\mathrm{d}x$.

The curve passes through the point with coordinates$\left(\frac{\pi }{3},\sqrt{3}\right)$.

Find an expression for$y$.

Answer:

Both of the standard integrals above are in the exam formula booklet.

With ln, it can sometimes be useful to write the constant of integration,$c$, as a logarithm. I.e., by letting $c=\ln k$ for some (positive) constant $k$.

Using the laws of logarithms, the answer can then be written as a single term. For example:

$\begin{array}{rcl}\int \frac{1}{x}\mathrm{d}x& =& \ln \left|x\right|+c\\ & =& \ln \left|x\right|+\ln k\\ & =& \ln \left(k\left|x\right|\right)\end{array}$

A curve has the gradient function$f^{\text{'}}\left(x\right)=\frac{3}{3x+2}+{\mathrm{e}}^{4-x}$.

Given that the exact value of$f\left(1\right)$ is$\ln 10-{\mathrm{e}}^3$, find an expression for$f\left(x\right)$.

Answer: