# Integration

## Integration Revision

**Integration**

**Integration** is the inverse of **differentiation**. It is represented by the symbol \int. **Indefinite** **integrals**, which we will look at on this page, have **multiple answers** because of the **constant of integration**. **Definite** **integrals** have **limits** and **do not have a**** constant of integration** – they are used to **find the area under a graph**.

The following topics build on the content in this page.

**Integration is the Inverse of Differentiation**

**Integration** is the inverse of **differentiation**. This is the **Fundamental Theorem of Calculus**.

**Example: **We know that the **derivative** of x^{2} is 2x. This means that the **integral** of 2x is x^{2}, which we write as \int2xdx=x^{2}. **Do not forget** the dx, it will be **important in later sections**.

However, this definition of** integration** **causes a problem**.

**Example: Differentiate**

i) x^{3}

ii) x^{3}+3

iii) x^{3}-19

You will notice that all of these are 3x^{2}. So what is \int3x^{2}dx? Is it x^{3}, x^{3}+3 or x^{3}-19? Under the rule established above** it is all three**.

This is where the **constant of integration** comes in.

In fact, we write \int3x^{2}dx=x^{3}+c, where c **represents any constant**, because all constants **differentiate** to 0. We do this for** all indefinite integrals**.

**Integrating a Polynomial**

To** integrate** something of the form x^{n}, **add** \mathbb{1} **to the power then divide by the new power**, to get \dfrac{1}{n+1}x^{n+1}

\int x^{n}dx=\dfrac{1}{n+1}x^{n+1}+c

As with **differentiation**, we can **integrate** **term by term**, and the **integral** of n times a function is n times the **integral** of a function.

This means that we can now **integrate** **polynomials**.

**Example: Integrate** x^{2}+3x+4

Do the **integration** **term by term**.

For x^{2}, we add 1 to the power to get 3, then divide by the new power, which is 3, to get \dfrac{1}{3}x^{3}

For x=x^{1}, we add 1 to the power to get 2, then divide by the new power, which is 2, to get \dfrac{1}{2}x^{2}

For 1=x^{0}, we add 1 to the power to get 1, then divide by the power, which is 1, to get x.

Putting it all together, and not forgetting +c:

\begin{aligned}\int x^{2}+3x+4dx&=\dfrac{1}{3}x^{3}+\left( 3\times\dfrac{1}{2}x^{2}\right) +\left( 4\times x\right) +c\\[1.2em]&=\dfrac{1}{3}x^{3}+\dfrac{3}{2}x^{2}+4x+c\end{aligned}**Note: **Our rule works for anything of the form x^{n}, **not just positive whole numbers**, so we can** integrate** far more than just polynomials. Also, it does not work for n=-1 because this would mean dividing by 0.

**Finding the Constant of Integration**

Sometimes you will be told that an **integral** **passes through some value**, and based on this asked to find the **constant of integration**.

**Example: **The curve y=f(x) passes through the point (8,200) and f'(x)=9x-5. Find f(x).

**Integrate** both sides,

f(8)=200

\left( \dfrac{9}{2}\times8^{2}\right) -\left( 5\times8\right) +c=200

\left( \dfrac{9}{2}\times64\right) -40+c=200

288-40+c=200

248+c=200

c=-48

f(x)=\dfrac{9}{2}x^{2}-5x-48

## Integration Example Questions

**Question 1: **Integrate:

i) x^{3}

ii) x^{4}

iii) 1

iv) x^{-2}

v) x^{\frac{1}{2}}

vi) x^{\frac{-3}{2}}

**[6 marks]**

i) \dfrac{1}{4}x^{4}+c

ii) \dfrac{1}{5}x^{5}+c

iii) x+c

iv) \dfrac{1}{-1}x^{-1}+c=\dfrac{-1}{x}+c

v) \dfrac{1}{\left( \dfrac{3}{2}\right) }x^{\frac{3}{2}}+c=\left( \dfrac{2}{3}x^{\frac{3}{2}}\right) +c

vi) \dfrac{1}{\left( \dfrac{-1}{2}\right) }x^{\frac{-1}{2}}+c=-2x^{\frac{-1}{2}}+c

**Question 2: **Integrate:

i) 3x^{2}

ii) \dfrac{1}{5}x^{6}

iii) 6x^{-4}

iv) 12x^{\frac{3}{4}}

**[4 marks]**

i) \left( 3\times\dfrac{1}{3}x^{3}\right) +c=x^{3}+c

ii) \left( \dfrac{1}{5}\times\dfrac{1}{7}x^{7}\right)+c=\dfrac{1}{35}x^{7}+c

iii) \left( 6\times\dfrac{1}{-3}x^{-3}\right) +c=-2x^{-3}+c

iv) \left( 12\times\dfrac{1}{\left( \dfrac{7}{4}\right) }x^{\frac{7}{4}}\right) +c=\dfrac{48}{7}x^{\frac{7}{4}}+c

**Question 3: **Integrate:

i) x^{2}+2x+1

ii) x^{4}+8x^{3}+2x^{2}+9x+6

iii) x^{\frac{1}{2}}+4x^{\frac{-5}{6}}

**[6 marks]**

i) \dfrac{1}{3}x^{3}+x^{2}+x+c

ii) \dfrac{1}{5}x^{5}+2x^{4}+\dfrac{2}{3}x^{3}+\dfrac{9}{2}x+6x+c

iii) \dfrac{2}{3}x^{\frac{3}{2}}+24x^{\frac{1}{6}}+c

**Question 4: **The graph of y=f(x) passes through the point (1,2), and f'(x)=3x+4. Find f(x).

**[3 marks]**

Integrate both sides.

f(x)=\dfrac{3}{2}x^{2}+4x+c

f(1)=2

2=\left( \dfrac{3}{2}\times1^{2}\right) +\left( 4\times1\right) +c

2=\dfrac{3}{2}+4+c

2=\dfrac{11}{2}+c

c=-\dfrac{7}{2}

f(x)=\dfrac{3}{2}x^{2}+4x-\dfrac{7}{2}

## Integration Worksheet and Example Questions

### Integration

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