Antiderivatives and the standard integrals

What you will be able to do

• say what an antiderivative is, and why the constant of integration is always there;
• write the standard integrals and use the rules for sums and constant multiples;
• find the one curve of the family that passes through a given point.

Going backwards

You know that the derivative of $x^2$ is $2x$. Now ask the question the other way. Which function has the derivative $2x$? The function $x^2$ does. So does $x^2 + 5$, and so does $x^2 - 3$. A constant disappears when we differentiate, so we cannot tell it from the derivative alone.

A function $F$ is an antiderivative of $f$ on an interval when $F'(x) = f(x)$ for every $x$ in the interval. Many texts also call $F$ a primitive of $f$. The whole family of antiderivatives is called the indefinite integral of $f$, and we write

The sign $\int$ is the integral sign, $f(x)$ is the integrand, and $dx$ names the variable. The letter $C$ is the constant of integration. It stands for any real number.

Why the constant is always there

Two antiderivatives of the same function on an interval differ by a constant. The reason is short. If $F' = f$ and $G' = f$, then $(F - G)' = 0$, so $F - G$ has slope zero everywhere on the interval. A function with zero slope everywhere is constant.

The picture shows the family for $f(x) = x^2 - 2x$. The curves have the same shape and sit at different heights. At the same value of $x$ their tangents are parallel.

The standard integrals

Each line below is a derivative from the last chapter, read backwards. You can check any of them by differentiating the right-hand side.

• $\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \ne -1$
• $\displaystyle\int \dfrac{1}{x}\,dx = \ln|x| + C$
• $\displaystyle\int e^x\,dx = e^x + C$ and $\displaystyle\int a^x\,dx = \dfrac{a^x}{\ln a} + C$
• $\displaystyle\int \sin x\,dx = -\cos x + C$ and $\displaystyle\int \cos x\,dx = \sin x + C$
• $\displaystyle\int \sec^2 x\,dx = \tan x + C$ and $\displaystyle\int \operatorname{cosec}^2 x\,dx = -\cot x + C$
• $\displaystyle\int \sec x\tan x\,dx = \sec x + C$ and $\displaystyle\int \operatorname{cosec} x\cot x\,dx = -\operatorname{cosec} x + C$
• $\displaystyle\int \dfrac{dx}{\sqrt{1 - x^2}} = \sin^{-1}x + C$ and $\displaystyle\int \dfrac{dx}{1 + x^2} = \tan^{-1}x + C$

The letter $a$ is a positive constant not equal to $1$. The absolute value in $\ln|x|$ matters: it makes the answer correct for negative $x$ too.

Two rules

Integration keeps sums and constant multiples, just as differentiation does.

There is no such rule for a product. The integral of $f \times g$ is not the product of the integrals. You will meet the correct tools in the next lessons and in the next course.

A worked example

Find $\displaystyle\int \left(6x^2 - 4x + 5\right)dx$.

1. Split the sum. Integrate each term on its own.
2. Use the power rule. $\int 6x^2\,dx = 6 \times \dfrac{x^3}{3} = 2x^3$. Next, $\int 4x\,dx = 4 \times \dfrac{x^2}{2} = 2x^2$. Last, $\int 5\,dx = 5x$.
3. Add one constant. The answer is $2x^3 - 2x^2 + 5x + C$. One $C$ is enough for the whole sum.
4. Check. The derivative is $6x^2 - 4x + 5$, which is the integrand.

One curve out of the family

A condition picks one curve. Suppose a curve has the slope $\dfrac{dy}{dx} = 3x^2 - 2x$ and passes through $(1, 4)$.

1. Integrate: $y = x^3 - x^2 + C$.
2. Put in the point: $4 = 1 - 1 + C$, so $C = 4$.
3. The curve is $y = x^3 - x^2 + 4$.

Practise with new numbers

Try it in Python

The program tests three candidates for an antiderivative of $f(x) = 3x^2 - 4x$. For each one it compares the slope of the candidate with $f$ at four points.

Common mistakes

• Forgetting the constant. Without $+ C$ the answer is only one curve of the family.
• Using the power rule for $\dfrac{1}{x}$. The power rule would divide by $0$. The integral of $\dfrac{1}{x}$ is $\ln|x|$.
• Integrating a product term by term. $\int x \cdot x\,dx$ is $\dfrac{x^3}{3}$, not $\dfrac{x^2}{2} \times \dfrac{x^2}{2}$.
• Skipping the check. Differentiate your answer. It takes ten seconds and catches most slips.

Remember these

• $F$ is an antiderivative of $f$ when $F' = f$. All antiderivatives differ by a constant.
• $\int f(x)\,dx = F(x) + C$ is the whole family. A point on the curve fixes $C$.
• The power rule is $\dfrac{x^{n+1}}{n+1}$ for $n \ne -1$, and $\int \dfrac{dx}{x} = \ln|x|$.
• Sums and constant multiples can be integrated term by term. Products cannot.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find $\displaystyle\int \left(4x^3 - 3x^2 + 2\right)dx$.

2. Find $\displaystyle\int \dfrac{(x + 1)^2}{x}\,dx$ for $x > 0$.

3. Find $\displaystyle\int \left(\sqrt{x} + \dfrac{1}{\sqrt{x}}\right)dx$ for $x > 0$.

4. A curve has the slope $2x - 3$ at every point and passes through $(2, 1)$. Find its equation.

5. True or false: $\displaystyle\int x \cdot x\,dx = \dfrac{x^2}{2} \times \dfrac{x^2}{2} + C$.

References

• OpenStax, Calculus Volume 1, Chapter 5, section 5.4 Integration Formulas and the Net Change Theorem.
• NCERT, Mathematics Part II for Class XII, Chapter 7, the sections on integration as the inverse process of differentiation.