In this article, we will be looking at some important properties of definite integrals which will be useful in evaluating such integrals effectively. We will also look at the proofs of each of these properties to gain a better understanding of them.

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## List of Properties of Definite Integrals

**1.Â **âˆ«_{a}^{b} f(x) dx = âˆ«_{a}^{b} f(t) dt

**2.**Â âˆ«_{a}^{b} f(x) dx = â€“ âˆ«_{b}^{a} f(x) dx â€¦ [Also, âˆ«_{a}^{a} f(x) dx = 0]

**3.**Â âˆ«_{a}^{b} f(x) dx = âˆ«_{a}^{c} f(x) dx + âˆ«_{c}^{b} f(x) dx

**4.**Â âˆ«_{a}^{b} f(x) dx = âˆ«_{a}^{b} f(a + b â€“ x) dx

**5.**Â âˆ«_{0}^{a} f(x) dx = âˆ«_{0}^{a} f(a â€“ x) dx â€¦ [this is derived from P04]

**6.**Â âˆ«_{0}^{2a} f(x) dx = âˆ«_{0}^{a} f(x) dx + âˆ«_{0}^{a} f(2a â€“ x) dx

**7.**Â Two parts

- âˆ«
_{0}^{2a}f(x) dx = 2 âˆ«_{0}^{a}f(x) dx â€¦ if f(2a â€“ x) = f(x). - âˆ«
_{0}^{2a}f(x) dx = 0 â€¦ if f(2a â€“ x) = â€“ f(x)

**8.**Â Two parts

- âˆ«
_{-a}^{a}f(x) dx = 2 âˆ«_{0}^{a}f(x) dx â€¦ if f(- x) = f(x) or it is an even function - âˆ«
_{-a}^{a}f(x) dx = 0 â€¦ if f(- x) = – f(x) or it is an odd function

## Proofs of Definite Integrals Properties

### Property 1: âˆ«_{a}^{b} f(x) dx = âˆ«_{a}^{b} f(t) dt

The proof for this property is not needed since simply by substituting x = t, the desired output is achieved.

**You can download Integrals Cheat Sheet by clicking on the download button below**

**Browse more Topics under Integrals**

- Fundamental Theorem of Calculus
- Introduction to Integration
- Properties of Indefinite IntegralsÂ
- Definite Integral as a Limit of a Sum
- Integration by Partial Fractions
- Integration by Parts
- Integration by Substitutions
- Integral of Some Particular Functions
- Integral of the Type e^x[f(x) + f'(x)]dx

### Property 2: âˆ«_{a}^{b} f(x) dx = â€“ âˆ«_{b}^{a} f(x) dx â€¦ [Also, âˆ«_{a}^{a} f(x) dx = 0]

Let I = âˆ«_{a}^{b} f(x) dx. If â€˜Fâ€™ is the anti-derivative of â€˜fâ€™, then by using the second fundamental theorem of calculus, we have I = F(b) â€“ F(a) = â€“ [F(a) â€“ F(b)] = â€“ âˆ«_{b}^{a} f(x) dx. Also, if a = b, then I = F(b) â€“ F(a) = F(a) â€“ F(a) = 0. Hence, âˆ«_{a}^{a} f(x) dx = 0.

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### Property 3: âˆ«_{a}^{b} f(x) dx = âˆ«_{a}^{c} f(x) dx + âˆ«_{c}^{b} f(x) dx

If â€˜Fâ€™ is the anti-derivative of â€˜fâ€™, then by using the second fundamental theorem of calculus, we have

- âˆ«
_{a}^{b}f(x) dx = F(b) â€“ F(a) â€¦ (1) - âˆ«
_{a}^{c}f(x) dx = F(c) â€“ F(a) â€¦ (2) - âˆ«
_{c}^{b}f(x) dx = F(b) â€“ F(c) â€¦ (3)

Adding equations (2) and (3), we get

âˆ«_{a}^{c} f(x) dx + âˆ«_{c}^{b} f(x) dx = F(c) â€“ F(a) + F(b) â€“ F(c)

= F(b) â€“ F(a) = âˆ«_{a}^{b} f(x) dx

### Property 4: âˆ«_{a}^{b} f(x) dx = âˆ«_{a}^{b} f(a + b â€“ x) dx

Let, t = (a + b â€“ x), or x = (a + b â€“ t), so that dt = â€“ dx â€¦ (4)

Also, observe that when x = a, t = b and when x = b, t = a. Hence, âˆ«_{a}^{b} will be replaced by âˆ«_{b}^{a} when we replace x by t. Therefore,

âˆ«_{a}^{b} f(x) dx = â€“ âˆ«_{b}^{a} f(a + b â€“ t) dt â€¦ from equation (4)

From Property 2, we know that âˆ«_{a}^{b} f(x) dx = â€“ âˆ«_{b}^{a} f(x) dx. Using this property, we get

âˆ«_{a}^{b} f(x) dx = âˆ«_{a}^{b} f(a + b â€“ t) dt

Next, using Property 1, we get

âˆ«_{a}^{b} f(x) dx = âˆ«_{a}^{b} f(a + b â€“ x) dx

### Property 5: âˆ«_{0}^{a} f(x) dx = âˆ«_{0}^{a} f(a â€“ x) dx

Let, t = (a â€“ x) or x = (a â€“ t), so that dt = â€“ dx â€¦ (5)

Also, observe that when x = 0, t = a and when x = a, t = 0. Hence, âˆ«_{0}^{a} will be replaced by âˆ«_{a}^{0} when we replace x by t. Therefore,

âˆ«_{0}^{a} f(x) dx = â€“ âˆ«_{a}^{0} f(a â€“ t) dt â€¦ from equation (5)

From Property 2, we know that âˆ«_{a}^{b} f(x) dx = â€“ âˆ«_{b}^{a} f(x) dx. Using this property, we get

âˆ«_{0}^{a} f(x) dx = âˆ«_{0}^{a} f(a â€“ t) dt

Next, using Property 1, we get

âˆ«_{0}^{a} f(x) dx = âˆ«_{0}^{a} f(a â€“ x) dx

### Property 6: âˆ«_{0}^{2a} f(x) dx = âˆ«_{0}^{a} f(x) dx + âˆ«_{0}^{a} f(2a â€“ x) dx

From Property 3, we know that

âˆ«_{a}^{b} f(x) dx = âˆ«_{a}^{c} f(x) dx + âˆ«_{c}^{b} f(x) dx

Therefore, âˆ«_{0}^{2a} f(x) dx = âˆ«_{0}^{a} f(x) dx + âˆ«_{a}^{2a} f(x) dx = I_{1} + I_{2} â€¦ (6)

Where, I_{1} = âˆ«_{0}^{a} f(x) dx and I_{2} = âˆ«_{a}^{2a} f(x) dx

Let, t = (2a â€“ x) or x = (2a â€“ t), so that dt = â€“ dx â€¦ (7)

Also, observe that when x = a, t = a, and when x = 2a, t = 0. Hence, âˆ«_{a}^{2a} will be replaced by âˆ«_{a}^{0} when we replace x by t. Therefore,

I_{2} = âˆ«_{a}^{2a} f(x) dx = â€“ âˆ«_{a}^{0} f(2a â€“ t) dt â€¦ from equation (7)

From Property 2, we know that âˆ«_{a}^{b} f(x) dx = â€“ âˆ«_{b}^{a} f(x) dx. Using this property, we get

I_{2} = âˆ«_{0}^{a} f(2a â€“ t) dt

Next, using Property 1, we get

I_{2} = âˆ«_{0}^{a} f(2a â€“ x) dx

Replacing the value of I_{2} in equation (6), we get

âˆ«_{0}^{2a} f(x) dx = âˆ«_{0}^{a} f(x) dx + âˆ«_{0}^{a} f(2a â€“ x) dx

### Property 7: âˆ«_{0}^{2a} f(x) dx = 2 âˆ«_{0}^{a} f(x) dx â€¦ if f(2a â€“ x) = f(x) and

âˆ«_{0}^{2a} f(x) dx = 0 â€¦ if f(2a â€“ x) = â€“ f(x)

From Property 5, we know that

âˆ«_{0}^{2a} f(x) dx = âˆ«_{0}^{a} f(x) dx + âˆ«_{0}^{a} f(2a â€“ x) dx â€¦ (8)

Now, if f(2a â€“ x) = f(x), then equation (8) becomes

âˆ«_{0}^{2a} f(x) dx = âˆ«_{0}^{a} f(x) dx + âˆ«_{0}^{a} f(x) dx

= 2 âˆ«_{0}^{a} f(x) dx

And, if f(2a â€“ x) = â€“ f(x), then equation (8) becomes

âˆ«_{0}^{2a} f(x) dx = âˆ«_{0}^{a} f(x) dx â€“ âˆ«_{0}^{a} f(x) dx = 0

### Property 8: âˆ«_{-a}^{a} f(x) dx = 2 âˆ«_{0}^{a} f(x) dx â€¦ if f(- x) = f(x) or it is an even function and âˆ«_{-a}^{a} f(x) dx = 0 â€¦ if f(- x) = – f(x) or it is an odd function

Using Property 3, we have

âˆ«_{-a}^{a} f(x) dx = âˆ«_{-a}^{0} f(x) dx + âˆ«_{0}^{a} f(x) dx = I_{1} + I_{2} â€¦ (9)

Where, I_{1} = âˆ«_{-a}^{0} f(x) dx I_{2} = âˆ«_{0}^{a} f(x) dx

**Consider I _{1}**

Let, t = â€“ x or x = â€“ t, so that dt = â€“ dx â€¦ (10)

Also, observe that when x = â€“ a, t = a, and when x = 0, t = 0. Hence, âˆ«_{-a}^{0} will be replaced by âˆ«_{a}^{0} when we replace x by t. Therefore,

I_{1} = âˆ«_{-a}^{0} f(x) dx = â€“ âˆ«_{a}^{0} f(â€“ t) dt â€¦ from equation (10)

From Property 2, we know that âˆ«_{a}^{b} f(x) dx = â€“ âˆ«_{b}^{a} f(x) dx. Using this property, we get

I_{1} = âˆ«_{-a}^{0} f(x) dx = âˆ«_{0}^{a} f(â€“ t) dt

Next, using Property 1, we get

I_{1} = âˆ«_{-a}^{0} f(x) dx = âˆ«_{0}^{a} f(â€“ x) dx

Replacing the value of I_{2} in equation (9), we get

âˆ«_{-a}^{a} f(x) dx = I_{1} + I_{2} = âˆ«_{0}^{a} f(â€“ x) dx + âˆ«_{0}^{a} f(x) dx â€¦ (11)

Now, if â€˜fâ€™ is an even function, then f(â€“ x) = f(x). Therefore, equation (11) becomes

âˆ«_{-a}^{a} f(x) dx = âˆ«_{0}^{a} f(x) dx + âˆ«_{0}^{a} f(x) dx = 2 âˆ«_{0}^{a} f(x) dx

And, if â€˜fâ€™ is an odd function, then f(â€“ x) = â€“ f(x). Therefore, equation (11) becomes

âˆ«_{-a}^{a} f(x) dx = â€“ âˆ«_{0}^{a} f(x) dx + âˆ«_{0}^{a} f(x) dx = 0

## Solved Examples on Definite Integrals

**Question 1: Evaluate âˆ« _{-1}^{2} |x^{3} â€“ x| dx**

**Answer :** Observe that, (x^{3} â€“ x) â‰¥ 0 on [â€“ 1, 0],Â (x^{3} â€“ x) â‰¤ 0 on [0, 1] andÂ (x^{3} â€“ x) â‰¥ 0 on [1, 2]

Hence, using Property 3, we can write

âˆ«_{-1}^{2} |x^{3} â€“ x| dx = âˆ«_{-1}^{0} (x^{3} â€“ x) dx + âˆ«_{0}^{1} â€“ (x^{3} â€“ x) dx + âˆ«_{1}^{2} (x^{3} â€“ x) dx

= âˆ«_{-1}^{0} (x^{3} â€“ x) dx + âˆ«_{0}^{1} (x â€“ x^{3}) dx + âˆ«_{1}^{2} (x^{3} â€“ x) dx

Solving the integrals, we get

âˆ«_{-1}^{2} |x^{3} â€“ x| dx = [(x^{4}/4 â€“ (x^{2}/2)]_{-1}^{0} + [(x^{2}/2 â€“ (x^{4}/4)]_{0}^{1} + [(x^{4}/4 â€“ (x^{2}/2)]_{1}^{2
}= â€“ [1/4 â€“ Â½] + [1/2 â€“ Â¼] + [4 â€“ 2] â€“ [1/4 â€“ Â½] = 11/4.

**Question 2: What is meant by definite integral?**

**Answer:** A definite integral refers to an integral with upper and lower limits. If it is restricted to exist on the real line, the definite integral is called by the name of Riemann integral.

**Question 3: Differentiate between indefinite and definite integral?**

**Answer: **A definite integral is characterized by upper and lower limits. Moreover, the reason why it is called definite is because it provides a definite answer at the end of the problem. Indefinite integral, in contrast, refers to a form of integration that is more general in nature. Furthermore, the interpretation of the indefinite integral is as the considered functionâ€™s anti-derivative.

**Question 4: Is it possible for definite integrals to be positive?**

**Answer:** Yes, it is possible for a definite integral to be positive. Integrals measure the area between the curve in question and the x-axis over a specified interval. Furthermore, if all of the area that is within the interval exists above the curve and below the x-axis then the result shall certainly be negative.

**Question 5: Do definite integrals require constant of integration?**

**Answer:** No, definite integrals have no requirement of a constant of integration.

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