5.1 Further Integration

Integrate

Use calculus to find the value of

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

            .

Use calculus to show that

            .

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  are all linear functions of .

Hence show that

Given that and that  show that

Given

and that the graph of   passes through the point  find .

Integrate

Use calculus to find the value of

Find an expression for given that

Given that

use calculus to find the exact value of

Find

Integrate

Find an expression for given that

Find an expression for  given that

Find an expression for given that

Use calculus to find the exact value of

Use the identity

to show that

Hence, or otherwise, find the integral

Show that

Given that the graph of   passes through the point  and that

Find 

Explain why  must be excluded from the domain of .

Show that

where  is a constant.

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.

Use calculus to find the value of

giving your answer in the form   ,  where  and  are prime 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  , where  are constants, and .

Show that

Hence, or otherwise, find an expression for  given that

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

Given that

and also that

find .

Given that  , show that