Integrating Trigonometric, Exponential & Reciprocal Functions (DP IB Analysis & Approaches (AA): HL): Revision Note

The standard integrals of$\sin x$ and $\cos x$ are both in the exam formula booklet.

The integral for ${\sec }^{2}x$ is not in the formula booklet. However the derivative result $f\left(x\right)=\tan x\Rightarrow f^{\text{'}}\left(x\right)={\sec }^{2}x$ is in the formula booklet, so that can be used 'the other way round' to deduce the antiderivative.

Make sure you are familiar with the calculus formulas that are in the exam formula booklet. Many standard integrals are included, but remember that you can also use the standard derivatives in the booklet to help you find antiderivatives.

Just remember to add '+c', the constant of integration, for any indefinite integrals you solve.

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

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

2. $\int {\sec }^2(3x-\frac{\pi }{3})\mathrm{d}x$

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