Integration

1. The diagram below shows the graphs of and .
   

Calculate the shaded area.

 Hint

1. Find the points of intersection of the curves by solving the equation .
2. Find the area between the curves by evaluating the integral of the difference of the functions between the points of intersection.

 Solutions

1. The points of intersection are found by solving the equation:

Rearranging gives:

Solve above, we get and .

2. The area between the curves is given by the integral:

Simplifying the integrand:

Calculating the integral:

Evaluating at the limits:

2. Find the following indefinite integrals, using any appropriate method.
   (a)

 Hint

Use the substitution .

 Solutions

Let , then . substitution into the original integral gives:

Calculating the integral:

substitution back to gives the final answer:

where is the constant of integration.

(b)

 Hint

The special case of integral by recognition: the reciprocal function: .

 Solutions

Use the reciprocal function rule, let , then , so we have
so

If you prefer substitution: let , then , so the integral becomes .

3. In this question you must show detailed reasoning
   Find the exact values of the following:
   (a) .

 Hint

Integration by substitution, Let

 Solutions

Let , hence and . Then

Therefore the exact value is

(b) .

 Hint

This can use reciprocal rule or substitution

 Solutions

use substitution method:
Let , so . When , ; when , . Thus

Therefore,

use reciprocal rule:
Let , then , so we have

Evaluating from to gives

4. Find .

 Hint

use “integration by parts”

 Solutions

Let and . Then and . By integration by parts:

Therefore,

5. Use a substitution to show that where and are rational numbers to be found.

 Hint

Let Then , so and
Alternatively, we can write , but both methods lead to the same answer.

 Solutions

Compute the integral:

Let . Then , so and . Hence

Integrating gives

Factor and substitute :

Therefore

with

Alternatively, we can write ,

Let . Then , so , and , while . Hence

Integrating gives

Factor and substitute back :

Therefore

with

6. The diagram shows part of the curve for .
   

R is the region above the -axis bounded by the curve, the axis and the line .
Show that the area of R is where and are rational numbers to be found.

 Hint

Find the point where the curve meets the x axis: solve , since x is larger then 0, we have , so . Then calculate the area using definite integral from 1 to e of the function . Use integration by parts to compute the integral.

 Solutions

The curve is . It meets the -axis when , i.e. at . Hence the region above the -axis bounded by the curve, the -axis and the line has area

Compute the integral by parts. Let and . Then and , so

Therefore

Simplifying,

Thus the area can be written as with

7. In this question you must show detailed reasoning If , evaluate , giving your answer in simplified logarithmic form.

 Hint

We compute the integral by partial fractions. Assume:

Then work out the values of and . Then integrate both parts (terms).

 Solutions

Assume

Multiplying both sides by gives

Equating coefficients yields the linear system

Subtracting the second equation from the first gives , so . Then . Thus

Now integrate termwise:

where absolute-value signs are unnecessary because and on .

Evaluate the bounds:

Therefore,