Methods of Integration Practice PDF

Preview

Summary

integration of inverse trig practice is included here...

Description

Met hodsofI nt egrat i on

Met hodsofI nt egrat i on 1.

Evaluate

2.

Evaluate

3.

Evaluate

4.

Evaluate

5.

Find the exact value of

©

(2012) All Rights Reserved

1 of 6

For more info, go to www.hscintheholidays.com.au

6.

Let

The graph of y = f (x) is sketched above.

7.

8.

©

(i)

Show that f (x) is an even function.

(ii)

Find the area enclosed by y = f (x), the x-axis, x = -2, and x =2

(i)

Use the substitution y = √x to find

(ii)

Use the substitution z = x to find another expression for

(iii)

Use the results of part (i) and (ii) to express sin-1 (2x – 1) in terms of sin-1 (√x) for 0 < x < 1.

Use the table of standard integrals to show that

(2012) All Rights Reserved

2 of 6

For more info, go to www.hscintheholidays.com.au

9.

Evaluate

, by using the substitution u = 2x + 1.

10.

Find

, using substitution u = In 2x. 11.

Evaluate

, using the substitution u = 1 + 5x. 12.

Evaluate

by using the substitution u = 2t – 1. 13.

Evaluate

using the substitution x = t2 + 1. 14.

©

Use the substitution u = 9 – x2 to find

(2012) All Rights Reserved

3 of 6

For more info, go to www.hscintheholidays.com.au

15.

Evaluate

, using the substitution u = 1 + x. 16.

Using the substitution u = ex, find

17.

Using the substitution u = 2x + 1, or otherwise, find

18.

Evaluate

19.

Evaluate

20.

The shaded region bounded by y = 3 sin x, the x-axis and the line x = is rooted about the x=axis to form a solid. Calculate the volume of the solid. ©

(2012) All Rights Reserved

4 of 6

For more info, go to www.hscintheholidays.com.au

21.

Find

22.

(i)

Show that

(ii)

The function g(x) is given by g(x) = 2 + cos x. The graph y = g(x) for ≤x ≤ is rotate about the x-axis. Find the volume of the solid generated. (You may use the result of part (i).)

23.

Evaluate

24.

Evaluate

25.

Using the table of standard integrals, find the exact value of

26.

By using the substitution x = sin t, or otherwise, evaluate

©

(2012) All Rights Reserved

5 of 6

For more info, go to www.hscintheholidays.com.au

27.

©

Use the substitution u = tan x to evaluate

(2012) All Rights Reserved

6 of 6

For more info, go to www.hscintheholidays.com.au...