5.1 Further Integration

Find the integral

Find an expression for  given that

Use calculus to evaluate

giving your answer as a single logarithm.

Integrate

Use calculus to find the value of

Use a suitable substitution to find the following

Use the substitution  to show that

where  is the constant of integration.

Show that, for (where  is an integer),

Use calculus and your result from part (a) to show that

Use calculus to find the exact value of

Find an expression for  given that

Find an expression for  given that

Show that

Hence, or otherwise, find an expression for

Write    in the form  where , and are all linear functions of .

Hence show that

Use calculus to show that

Use integration by parts to find

Show that

where  is the constant of integration.

Express

as partial fractions.

Hence, or otherwise, find

Show that

Given that and that   show that

Given

and that the graph of    passes through the point (1, ln 144 ) find .

Find the integral

Use calculus to evaluate

Find an expression for  given that

Integrate

Use calculus to find the value of

Find an expression for  given that

Use a suitable substitution to find

Find an expression for  given that

Find an expression for  given that

Use calculus to find the exact value of

Use calculus and the substitution  to show that

Given that

use calculus to find the exact value of

Find

Use the identity 

to show that

Hence, or otherwise, find the integral

Show that

Use integration by parts to find, in terms of , the exact value of

Show that

can be written in the form

Hence find

writing your answer as a single logarithm.

Use calculus to find the value of

giving your answer to three significant figures.

Given that the graph of   passes through the point (0, 4) and that

Find

Explain why   must be excluded from the domain of 

Show that

where  is a constant.

Find an expression for given that

Integrate

Find an expression for  given that

Integrate

Use calculus and the substitution     to find the exact value of

Find

Show that

where is a constant, and  is the constant of integration.

Use calculus to find the exact value of

writing your answer in the form   , where and  are rational numbers to be found.

Show that

Hence, or otherwise, find the exact value of

Show that

where and are constants, and 

Using your working from part (a), or otherwise, evaluate  

giving your answer in terms of  and , where and are constants, and .

Show that

Hence, or otherwise, find an expression for  given that

Find

Find

Find the integral

Use integration by parts to show that

where  is the constant of integration.

Show that there are no positive values of and  that satisfy the equation

Given that

and also that 

find .

Express   in partial fractions.

Use your answer from part (a) to help find

Given that and  ,  show that