Integration is the opposite process to differentiation.

The **indefinite integral** of a function $f(x)$ with respect to $x$ is denoted by: \[\int f(x) \mathrm{d} x.\] The function appearing inside the integral, $f(x)$, is known as the **integrand**. We can find the indefinite integral of a function using the rules for finding the indefinite integral. It is important to note that the indefinite integral of a function is a function itself. We denote this function by $F(x)$, so we can write \[\int f(x) \mathrm{d} x = F(x) + C\] where $C$ is called the constant of integration and arises from the constant rule of differentiation (see here for more information about why it is necessary). Since integration is the *reverse* of differentiation, we have: \[F'(x)=f(x)\]

**Note**: Because integration is the opposite process of differentiation, the indefinite integral $F(x)$ is also referred to as an **antiderivative**.

The **definite integral** of a function $f(x)$: \[\int_{\large{a} }^{\large{b} }\,f(x)\, \mathrm{d}x\]

is a *number*. This number is equal to the area between the curve of the function and the $x$-axis and between $2$ specified values of $x$. For example, the integral above is equal to the area under the curve of $f(x)$ between the points $x=a$ and $x=b$.

We call $b$ the **upper limit** of integration and $a$ the **lower limit** of integration.

We can see that the definite integral of the function $f(x)$ is just the indefinite integral of $f(x)$ evaluated between $2$ values of $x$. We have: \[\int_{\large{a} }^{\large{b} }\,f(x)\, \mathrm{d}x=\Bigl[F(x)\Bigl]_{\large{a} }^{\large{b} }=F(b)-F(a).\] where $F(x)$ is the indefinite integral of $f(x)$.

**Note**: When evaluating definite integrals it is not necessary to include the constant of integration, $C$, that arises in indefinite integration.

Since integration is the opposite process of differentiation, the rules of integration are the rules of differentiation, reversed.

See the constant rule of differentiation. \[\int a\; \mathrm{d} x=ax+C\] where $a$ is a *non-zero* constant.

Find the indefinite integral of the function $f(y)=5$.

\[\int 5\; \mathrm{d} y=5y+C\]

See the power rule of differentiation.

\[\int x^n \mathrm{d} x=\dfrac{x^{n+1}}{n+1}+C\]

In words this says “add one to the exponent and divide by the new exponent”.

Find the indefinite integral of the function $f(x)=x^4$.

\[\int x^4 \mathrm{d} x=\frac{x^5}{5}+C\]

Find the indefinite integral of the function $f(x)=x^{~-\dfrac{3}{4}~}$.

\begin{align} \int x^{~-\frac{3}{4}~} \mathrm{d} x&=\frac{x^{~\left(-\frac{3}{4}+1\right)}~}{-\frac{3}{4}+1}+C\\ &=\frac{x^{~\frac{1}{4}~} }{\frac{1}{4} }+C\\ &=4x^{~\frac{1}{4}~}+C \end{align}

See the power rule of differentiation.

\[\int af(x) \mathrm{d} x= a\int f(x) \mathrm{d} x=aF(x)+C\] where $a$ is a non-zero constant.

In words this says that we can bring multiplicative constants ($a$ in this case) *outside* of the integral. We then integrate the function $f(x)$, multiply the result by the multiplicative constant, and add the constant of integration.

Find the indefinite integral of the function $f(x)=11x^3$.

\begin{align} \int 11x^3 \mathrm{d}&=11\int x^3 \mathrm{d} x\\ &=\dfrac{11}{4}x^4+C \end{align}

See the sum or difference rule of differentiation.

To integrate a sum (or difference) of terms, integrate each term separately and add (or subtract) the integrals.

**Note**: Remember to include only *one* constant of integration when you add or subtract the integrals.

Find the indefinite integral of the function $f(x)=2x^3+x^2-7$.

Using the multiplicative constant rule, the indefinite integral of the first term is: \begin{align}\int 2x^3 \mathrm{d} x&=\frac{2}{4}x^4+C\\ &=\frac{x^4}{2}+C \end{align}

We can then use the power rule to integrate the second term: \[\int x^2 \mathrm{d} x=\frac{x^3}{3}+C\] Finally, using the constant rule we can integrate the last term: \[\int 7 \mathrm{d} x=7x+C\] Adding the integrals of the first two terms and subtracting the integral of the last term gives: \[\int 2x^3+x^2-7 \mathrm{d} x=\frac{x^4}{2}+\frac{x^3}{3}-7x+C\]

The function of a function rule of integration is the opposite of the chain rule of differentiation.

Here we will consider the simple case where the function which has been differentiated was of the form $y=f(x)^n$. This is a function of a function because $y$ is a function of $f$ and $f$ is a function of $x$.

Using the chain rule, we have: \begin{align} \frac{dy}{dx}&=\frac{dy}{df}\times \frac{df}{dx}\\ &=n[f(x)]^{n-1}\times f'(x) \end{align}

Now, suppose we are asked to find the indefinite integral of a function of the form $n[f(x)]^{n-1}\times f'(x)$. By reversing the chain rule, we can see that the indefinite integral of this function is: \[\int n[f(x)]^{n-1}\times f'(x) \mathrm{d} x=f(x)^n+C\]

Find the indefinite integral of the function $g(x)=2x(x^2+1)$.

Since $2x$ is the derivative of $(x^2+1)$ we can apply the function of a function rule to integrate this function. We have $f'(x)=2x$, $f(x)=(x^2+1)$ and $f(x)^n=(x^2+1)$ since $n=1$ so: \begin{align} \int 2x(x^2+1) \mathrm{d} x&=\int n[f(x)]^{n-1} f'(x) \mathrm{d} x\\ &=f(x)^n+C\\ &=(x^2+1)+C \end{align}

Find the indefinite integral of the function $h(x)=15x^2(x^3+4)^5$.

Since $15x^2$ is the derivative of $(x^3+4)^5$ we can apply the function of a function rule to integrate this function. We have $f(x)=x^3+4$ and $f(x)^n=(x^3+4)^5$ so: \begin{align} \int 15x^2(x^3+4)^4 \mathrm{d} x&=\int n[f(x)]^{n-1} f'(x) \mathrm{d} x\\ &=f(x)^n+C\\ &=(x^3+4)^5+C \end{align}

Reversing the rules for differentiating the exponential function, we have: \[\int e^x \mathrm{d} x= e^x\] and \[\int g'(x) e^{g(x)}= e^{g(x)}\]

Find the indefinite integral of the function $f(x)=12x^2 e^{4x^3}$.

Here the power to which $ e$ is raised is a function of $x$ so we have $g(x)=4x^3$ and $g'(x)=12x^2$ so: \[\int 12x^2 e^{4x^3} \mathrm{d} x= e^{4x^3}\]

Suppose we want to find the area under the curve of the function $y=4{x^3}$ between $x=0$ and $x=4$. To evaluate a definite integral of any function first find the indefinite integral of the function. For our chosen function this is: \[\int f(x) \mathrm{d} x.=x^4+C\] We must then evaluate this function at the upper and lower limits, $x=4$ and $x=0$ respectively: \begin{align} \int_0^4 \, 4x^3 \, \mathrm{d}x&=\left[ x^4 \right]_{x=4}-\left[ x^4 \right]_{x=0}\\ &=256-0\\ &=256 \end{align} The area under the curve $y=4x^3$ between $x=0$ and $x=4$ is therefore equal to $256$. `

By definition, a firm’s marginal revenue ($MR$) function can be found by differentiating the firm’s total revenue ($TR$) function. Since integration is the reverse of differentiation, given a MR function, we can obtain the corresponding TR function by finding the indefinite integral of the marginal revenue function. We can use this same method to obtain the total cost function given a firm’s marginal cost function.

Suppose that a bakery’s $MR$ function is $MR=3q^2+2$, where $q$ is the quantity of bread loaves produced by the bakery and the revenue is in $£$. The bakery earns $£8,040$ in revenue from selling $20$ loaves of bread. What is bakery’s total revenue from producing $100$ loaves of bread?

We must first find the bakery’s total revenue function by finding the indefinite integral of the marginal revenue function: \begin{align} TR(q)&=\int MR(q) \mathrm{d} q\\ &=\int 3q^2+2 \mathrm{d} q\\ &=\dfrac{3q^3}{3}+2q+C\\ &=q^3+2q+C \end{align}

Now, we have been told that by producing $20$ loaves of bread, the bakery earns $£8,040$ in revenue. We can use this piece of information to determine the value of the constant of integration, and thus obtain the bakery's MR function. \begin{align} 8,040&=20^3+2\times 20+C\\ \Rightarrow C&=8,040-20^3+2\times 20\\ \Rightarrow C&=8,040-20^3+2\times 20\\ \Rightarrow C&=4 \end{align} so the total revenue function is: \[TR(q)=q^3+2q+4\] We can check that the total revenue is correct by differentiating it. The derivative should be equal to the given marginal revenue function: \[TR'(q)=3q^2+2=MR(q)\]

We can now use the total revenue function to determine the revenue earned by the bakery when it sells $100$ loaves of bread: \begin{align} TR(100)&=3\times 100^2+2\\ &=£30,002 \end{align}