**Differentiating tan ^{-1}x**

If y = tan^{-1}x, then x=tany.

We can differentiate both sides of this equation with respect to x (using implicit differentiation on the RHS) and then use Pythagoras’ Theorem on the result to get:

**Worked Examples**

Find in terms of x for each of the following:

- y=tan
^{-1}2x; - y=tan
^{-1}3x; - y=tan
^{-1}(x/2); - y=tan
^{-1}(x/3).

**Exercise**

**Answers**

**Integration of **

Based on the derivative of tan^{-1}x, we can see that:

If the x^{2} has a coefficient, it is convenient to start by “moving it outside” the integral sign, eg:

**Worked Example**

1.)

2.)

3.) Find the indefinite integral of

**Exercise**

**Answers**

**Integration of **

As with other examples, we may sometimes need to modify an expression slightly to get it into this form, for example taking a constant outside of the integral sign.

**Worked Examples**

1.)

2.)

3.)

4.)

5.)

**Exercise**

**Answers**

**Integration by Substitution**

This is a very powerful method to simplify an integration.

The integral **and** the limitsmust be changed to the new variable.

We must change back to the original variable at the end.

**Worked Examples**

1.) FInd the following indefinite integrals by making the suggested substitution. Remember to give your final answer in terms of x:

2.) Find the area of the shaded region for each of the following graphs:

3.) Use the substitution u = sinx + 1 to find

4.) Prove that

**Exercise**

**Answers**

**Using Partial Fractions in Integration**

We can use partial fractions to make integration easier.

1.) Use partial fractions to find the following integrals:

**Exercise**

**Answers**

**Integration by Parts**

If an integral is a product of something easy to integrate and something difficult to integrate, we can use **integration by parts** to make it easier to integrate.

The integration by parts formula is derived by rearranging the product rule.

**Integration by Parts Formula**

**Worked Examples**

1.) Find

2.) Find

Sometimes we need to use the technique more than once.

3.) Find

**Exercise**

**Mixed Exercise**

**Answers**

**Mixed Exercise Answers**