# Functions and algebra: Analyse and represent mathematical and contextual situations using integrals and find areas under curves by using integration rules

### Subject outcome

Subject outcome 2.5: Analyse and represent mathematical and contextual situations using integrals and find areas under curves by using integration rules

### Learning outcomes

• Find the integrals of the following:
$\scriptsize \int a{{x}^{n}}dx$ ; $\scriptsize \int \displaystyle \frac{a}{x}dx$ ; $\scriptsize \int a{{e}^{{kx}}}dx$ ; $\scriptsize \int a\text{sin}kxdx$ ; $\scriptsize \int a\text{cos}kxdx$ ; $\scriptsize \int \text{ase}{{\text{c}}^{2}}kxdx$
Where:
$\scriptsize \int a{{x}^{n}}dx=\displaystyle \frac{{a{{x}^{{n+1}}}}}{{n+1}}+c$
$\scriptsize \int \displaystyle \frac{a}{x}dx=a\ln x+c$
$\scriptsize \int a{{e}^{{kx}}}dx=\displaystyle \frac{{a{{e}^{{kx}}}}}{k}+c$
$\scriptsize \int a\text{sin}kxdx=\displaystyle \frac{{-a\text{cos}kx}}{k}+c$
$\scriptsize \int a\text{cos}kxdx=\displaystyle \frac{{a\text{sin}kx}}{k}+c$
Note:

• Simplifications may be required where necessary.
• Integrals of polynomials may be assessed.
• Integration by parts is excluded.
• Use the upper and lower limits to calculate definite integrals.
• Determine the area under a curve by:
• Working from a given graph or by sketching a graph.
• Working with an area bounded by a curve, the x-axis, an upper and a lower limit.
• Splitting the area into two intervals when the graph crosses the x-axis.
Note:

• Integrals with respect to the x-axis only.
• Areas between two curves are excluded.
• The y-axis (x = 0) may be used as an upper or lower limit.

### Unit 1 outcomes

By the end of this unit you will be able to:

• Understand the relationship between differentiation and integration (anti-derivative).
• Define integration as the approximate area under curves.

### Unit 2 outcomes

By the end of this unit you will be able to:

• Apply the rules of integration to various functions.
• Calculate the definite integral.

### Unit 3 outcomes

By the end of this unit you will be able to:

• Find the area under a curve between two points and bound by the x-axis.