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Chapter 7 · Integrals

Exercise 7.1 · Q8: ∫(ax2+bx+c) dx\int (ax^2 + bx + c)\,dx

Exercise 7.1Properties of indefinite integrals · Section 7.2.1
The question

Find the following integral:

∫(ax2+bx+c)dx\int\left(ax^2 + bx + c\right)dx
Exam answer

Let I=∫(ax2+bx+c)dxI = \displaystyle\int\left(ax^2 + bx + c\right)dx⁠.

I=a∫x2 dx+b∫x dx+c∫1 dx\begin{aligned} I &= a\int x^2\,dx + b\int x\,dx \\ &\quad + c\int 1\,dx \end{aligned}

∴ I=ax33+bx22+cx+C\therefore\ I = \dfrac{ax^3}{3} + \dfrac{bx^2}{2} + cx + C⁠.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

A quadratic is a sum of constant multiples of powers of xx⁠: a⋅x2+b⋅x1+c⋅x0a\cdot x^2 + b\cdot x^1 + c\cdot x^0⁠. Integrate each power with the power rule (raise the power by one, divide by the new power) and carry the constants a,b,ca, b, c along unchanged.

Given, to find, and the plan
Given
The integrand ax2+bx+cax^2 + bx + c⁠, where a,b,ca, b, c are fixed real constants; it is defined for every real xx⁠.
To find
The indefinite integral.
Plan
(1) Split into three integrals and take out the constants (Property (V)). (2) Apply the power rule to each. (3) Assemble with one constant of integration, named CC (capital) to keep it apart from the given cc⁠. (4) Differentiate back to check.

Basics you need

1Power rule for integrals

∫xn dx=xn+1n+1+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C for n≠−1n \ne -1⁠; in particular ∫dx=x+C\int dx = x + C (item (i), Section 7.2). Basics: powers of x and the power rule

2Term by term, constants outside

∫(k1f1+k2f2+k3f3) dx=k1∫f1+k2∫f2+k3∫f3\int (k_1f_1 + k_2f_2 + k_3f_3)\,dx = k_1\int f_1 + k_2\int f_2 + k_3\int f_3 (Property (V), Section 7.2.1). Lesson: integration as anti differentiation

Solution

  1. Split and take out the constants

    ∫(ax2+bx+c)dx=a∫x2 dx+b∫x dx+c∫dx.\begin{aligned} &\int\left(ax^2 + bx + c\right)dx \\ &= a\int x^2\,dx + b\int x\,dx + c\int dx. \end{aligned}
    aa⁠, bb and cc do not depend on xx⁠, so they behave exactly like the numbers 22 or 55 would: Property (V) lets us pull each one outside its integral. What remains are three bare powers of xx⁠.
    Property (V), Section 7.2.1. Lesson: integration as anti differentiation

  2. Power rule on each piece

    ∫x2 dx=x33,∫x dx=x22,∫dx=x,\begin{aligned} \int x^2\,dx &= \frac{x^{3}}{3}, \\ \int x\,dx &= \frac{x^{2}}{2}, \\ \int dx &= x, \end{aligned}

    each up to a constant of integration.

    For x2x^2 take n=2n = 2⁠: the power becomes 33 and we divide by 33⁠. For x=x1x = x^1 take n=1n = 1⁠: the power becomes 22 and we divide by 22⁠. The constant function 11 has anti derivative xx⁠.
    Standard formulae, item (i), Section 7.2. Basics: the power rule

  3. Assemble with one constant CC

    ∫(ax2+bx+c)dx=ax33+bx22+cx+C.\int\left(ax^2 + bx + c\right)dx = \frac{ax^3}{3} + \frac{bx^2}{2} + cx + C.
    The small cc is part of the question, a fixed number; the capital CC is the arbitrary constant of integration. They are different things, and writing them with different letters keeps the answer readable.
    Note after Example 2: one constant of integration in the final answer. Example 2

  4. Check: differentiate the answer back

    ddx(ax33+bx22+cx+C)=3ax23+2bx2+c+0=ax2+bx+c.  ✓\begin{aligned} &\frac{d}{dx}\left(\frac{ax^3}{3} + \frac{bx^2}{2} + cx + C\right) \\ &= \frac{3ax^2}{3} + \frac{2bx}{2} + c + 0 \\ &= ax^2 + bx + c. \;\checkmark \end{aligned}
    Each term returns to its place in the integrand, for every real xx and for every choice of the constants a,b,ca, b, c⁠.
    Property (I), Section 7.2.1. Basics: integration undoes differentiation
Result ∫(ax2+bx+c)dx=ax33+bx22+cx+C,\int\left(ax^2 + bx + c\right)dx = \frac{ax^3}{3} + \frac{bx^2}{2} + cx + C,

valid for every real xx⁠. (Textbook answer: ax33+bx22+cx+C\frac{ax^3}{3} + \frac{bx^2}{2} + cx + C⁠.)

Check it with numbers

Take a=1a = 1⁠, b=2b = 2⁠, c=3c = 3⁠, so F(x)=x33+x2+3xF(x) = \frac{x^3}{3} + x^2 + 3x and the integrand is x2+2x+3x^2 + 2x + 3⁠. At x=1x = 1 with h=0.001h = 0.001⁠:

F(1.001)−F(0.999)0.002=4.339335333667−4.3273353330000.002≈6.0000003.\begin{aligned} &\frac{F(1.001) - F(0.999)}{0.002} \\ &= \frac{4.339335333667 - 4.327335333000}{0.002} \\ &\approx 6.0000003. \end{aligned}

The integrand at x=1x = 1 is 1+2+3=61 + 2 + 3 = 6⁠. ✓

Common mistakes
  1. Integrating cc to 00 or to cc⁠. A constant integrates to the constant times xx⁠: ∫c dx=cx\int c\,dx = cx⁠.
  2. Mixing up cc and CC⁠. Writing "+ c+\,c" at the end would confuse the given constant with the constant of integration.
  3. Treating aa or bb as variables. They are fixed numbers; they are not raised to any power and are not divided by anything new.
What to take away
  1. Any polynomial is integrated term by term: ∫∑akxk dx=∑akxk+1k+1+C\int \sum a_kx^k\,dx = \sum \frac{a_k x^{k+1}}{k+1} + C⁠.
  2. Letters standing for given constants are carried along like numbers.

Every fact used, and where it comes from

FactUsed inSource
Integrate term by term, constants outsideStep 1Property (V), Section 7.2.1
∫xn dx=xn+1n+1+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C⁠, n≠−1n \ne -1⁠; ∫dx=x+C\int dx = x + CStep 2Standard formulae, item (i), Section 7.2
One constant of integrationStep 3Note after Example 2, Section 7.2.1
Differentiation and integration undo each otherStep 4Property (I), Section 7.2.1
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