Substitution

Red Question: Evaluate

STEP 2005, Math III, #7: Show that if

,

then

where .

Find:

(i)

(ii)

STEP 2005, Math I, #5: (i) Evaluate the integral

in the cases and .

Deduce that

when

.

(ii) Evaluate the integral

in the different cases that arise according to the value of

.

STEP 2013, Math III, #1: Given that

show that

and

.

Hence show that

Let

By considering , or otherwise, evaluate .

STEP 2013, Math II, #2: For

, let

(i) For , show by means of a substitution that

and deduce that

Show also, for

and hence that

(ii) When

is a positive integer, show that

(iii) Use the substitution to show that

and evaluate

.

STEP 2004, Math I, #4: Differentiate

with respect to .

(i) Use the substitution

to show that

(ii) Determine

(iii) Determine

STEP 2008, Math II, #5: Evaluate the integrals

Show, using the binomial expansion, that . Show also that . Deduce that . Use this result to determine which of the above integrals is greater.

STEP 2006, Math I, #5: (i) Use the substitution to show that, for ,

where

is a constant.

(ii) Show that

STEP 1999, Math II, #6: Find if

(*)

By using changes of variable of the form (*), or otherwise, show that

and evaluate the integrals

and

STEP 2000, Math III, #2: Use the substitution

to evaluate the integral

Show that, for

,

where and .

STEP 2001, Math II, #6: Show that

Determine the values of

(i)

(ii)

STEP 2009, Math I #6: (i) Show that, for

,

(ii) Show by means of a substitution that

(iii) Evaluate:

(a)

(b)

STEP 2010, Math I, #4: Use the substitution

where

, to show that, for ,

Note: You may use without proof the result

.

The section of the curve

and is rotated through about the x-axis. Show that the volume enclosed is .

STEP 2013, Math I, #4: Show that, for

,

and

Evaluate the following integrals:

and

STEP 2010, Math II, #4: (i) Let

Use substitution to show that

and hence evaluate

in terms of .

Use this result to evaluate the integrals

and

(ii) Evaluate

STEP 2002, Math II, #1: Show that

By using the substitution

, or otherwise, show that

Hence evaluate the integral

STEP 2006, Math II, #4: By making the substitution

, show that

where is a given function of .

Evaluate the following integrals:

(i)

(ii)

(iii)

2008, Math I, #6: The function is defined by

is the inverse function to , so that . Sketch and on the same axes.

Verify, by evaluating each integral, that

where

and explain this result by means of a diagram.

STEP 2000, Math II, #6: Show that

where .

and deduce a similar result for

STEP 2011, Math III, #6: The definite integrals

and are defined by

Show, without evaluating any of them, that

and are all equal.