Definite integrals of polynomials

• A polynomial is an algebraic expression comprising variables, coefficients, and exponents.
• The definite integral of a polynomial can be evaluated using the power rule.
• The power rule states that if we have a polynomial of the form (f(x) = ax^n), where (a) and (n) are constants, then the definite integral of (f(x)) is given by: [\int_{a}^{b} f(x)dx = \frac{a}{n+1}(b^{n+1} - a^{n+1})]
• Example 1: Let’s evaluate the definite integral of (f(x) = 2x^3 - 3x^2 + 5x - 2) from 0 to 2 using the power rule. [\int_{0}^{2} f(x)dx = \int_{0}^{2} (2x^3 - 3x^2 + 5x - 2)dx] [= \frac{2}{4}(2^4 - 0^4) - \frac{3}{3}(2^3 - 0^3) + \frac{5}{2}(2^2 - 0^2) - 2(2 - 0)]

Definite integrals of exponential functions

• Exponential functions have the form (f(x) = a^x), where (a) is a constant.
• The definite integral of an exponential function can be evaluated using a logarithmic substitution.
• The logarithmic substitution involves substituting (u = \ln(a)x) and (du = \frac{\ln(a)}{x}dx).
• Example 1: Let’s evaluate the definite integral of (f(x) = e^x) from 0 to 1 using a logarithmic substitution. [\int_{0}^{1} f(x)dx = \int_{0}^{1} e^xdx] Let (u = \ln(e)x = x) and (du = dx). [\int_{0}^{1} e^xdx = \int_{0}^{1} e^udu = \int_{0}^{1} e^udu] [= [e^u]_{0}^{1} = e^1 - e^0 = e - 1]

Definite integrals of trigonometric functions

• Trigonometric functions, such as sine, cosine, and tangent, are commonly encountered in definite integrals.
• The definite integral of trigonometric functions can be evaluated using various trigonometric identities and properties.
• Example 1: Let’s evaluate the definite integral of (f(x) = \sin(x)) from 0 to (\pi) using trigonometric identities. [\int_{0}^{\pi} f(x)dx = \int_{0}^{\pi} \sin(x)dx] [= [-\cos(x)]_{0}^{\pi} = -(\cos(\pi) - \cos(0)) = -(-1 - 1) = 2]
• Example 2: Let’s evaluate the definite integral of (f(x) = \cos(x)) from 0 to (\frac{\pi}{2}) using trigonometric identities. [\int_{0}^{\frac{\pi}{2}} f(x)dx = \int_{0}^{\frac{\pi}{2}} \cos(x)dx] [= [\sin(x)]_{0}^{\frac{\pi}{2}} = (\sin(\frac{\pi}{2}) - \sin(0)) = (1 - 0) = 1]

Definite integrals of logarithmic functions

• Logarithmic functions have the form (f(x) = \ln(x)), where (x) is a positive real number.
• The definite integral of a logarithmic function can be evaluated using integration by parts.
• Integration by parts involves choosing one part of the integrand as the “first” part and differentiating it, while integrating the remaining part as the “second” part.
• Example 1: Let’s evaluate the definite integral of (f(x) = \ln(x)) from 1 to 2 using integration by parts. [\int_{1}^{2} f(x)dx = \int_{1}^{2} \ln(x)dx] Let (u = \ln(x)) and (dv = dx). [du = \frac{1}{x}dx \quad \text{and} \quad v = x] Using the integration by parts formula: [\int u dv = uv - \int v du] [\int_{1}^{2} \ln(x)dx = [x\ln(x)]{1}^{2} - \int{1}^{2}x\left(\frac{1}{x}dx\right)] [= \left[2\ln(2) - 1\ln(1)\right] - \left[\int_{1}^{2} dx\right]] [= 2\ln(2) - 1 - (2 - 1)]

Summary

• In this lecture, we covered four special functions of definite integrals:
  1. Definite integrals of polynomials
  2. Definite integrals of exponential functions
  3. Definite integrals of trigonometric functions
  4. Definite integrals of logarithmic functions
• Special techniques such as power rule, logarithmic substitution, trigonometric identities, and integration by parts were used to evaluate these special functions.
• Understanding and applying these techniques will help you solve complex definite integrals with ease.

Definite Integral - Special function

Slide 11

• In this topic, we will discuss special functions of definite integrals.
• Special functions are those integrals that have specific properties or require special techniques to evaluate.
• We will focus on four special functions:
  1. Definite integrals of polynomials
  2. Definite integrals of exponential functions
  3. Definite integrals of trigonometric functions
  4. Definite integrals of logarithmic functions

Slide 12

• Let’s start with the first special function: Definite integrals of polynomials.
• A polynomial is an algebraic expression comprising variables, coefficients, and exponents.
• The definite integral of a polynomial can be evaluated using the power rule.
• The power rule states that if we have a polynomial of the form (f(x) = ax^n), where (a) and (n) are constants, then the definite integral of (f(x)) is given by: [\int_{a}^{b} f(x)dx = \frac{a}{n+1}(b^{n+1} - a^{n+1})]

Slide 13

• Example 1: Let’s evaluate the definite integral of (f(x) = 2x^3 - 3x^2 + 5x - 2) from 0 to 2 using the power rule. [\int_{0}^{2} f(x)dx = \int_{0}^{2} (2x^3 - 3x^2 + 5x - 2)dx] [= \frac{2}{4}(2^4 - 0^4) - \frac{3}{3}(2^3 - 0^3) + \frac{5}{2}(2^2 - 0^2) - 2(2 - 0)]

Slide 14

• Let’s move on to the second special function: Definite integrals of exponential functions.
• Exponential functions have the form (f(x) = a^x), where (a) is a constant.
• The definite integral of an exponential function can be evaluated using a logarithmic substitution.
• The logarithmic substitution involves substituting (u = \ln(a)x) and (du = \frac{\ln(a)}{x}dx).

Slide 15

• Example 1: Let’s evaluate the definite integral of (f(x) = e^x) from 0 to 1 using a logarithmic substitution. [\int_{0}^{1} f(x)dx = \int_{0}^{1} e^xdx] Let (u = \ln(e)x = x) and (du = dx). [\int_{0}^{1} e^xdx = \int_{0}^{1} e^udu = \int_{0}^{1} e^udu] [= [e^u]_{0}^{1} = e^1 - e^0 = e - 1]

Slide 16

• Let’s now discuss the third special function: Definite integrals of trigonometric functions.
• Trigonometric functions, such as sine, cosine, and tangent, are commonly encountered in definite integrals.
• The definite integral of trigonometric functions can be evaluated using various trigonometric identities and properties.

Slide 17

• Example 1: Let’s evaluate the definite integral of (f(x) = \sin(x)) from 0 to (\pi) using trigonometric identities. [\int_{0}^{\pi} f(x)dx = \int_{0}^{\pi} \sin(x)dx] [= [-\cos(x)]_{0}^{\pi} = -(\cos(\pi) - \cos(0)) = -(-1 - 1) = 2]

Slide 18

• Example 2: Let’s evaluate the definite integral of (f(x) = \cos(x)) from 0 to (\frac{\pi}{2}) using trigonometric identities. [\int_{0}^{\frac{\pi}{2}} f(x)dx = \int_{0}^{\frac{\pi}{2}} \cos(x)dx] [= [\sin(x)]_{0}^{\frac{\pi}{2}} = (\sin(\frac{\pi}{2}) - \sin(0)) = (1 - 0) = 1]

Slide 19

• Let’s move on to the last special function: Definite integrals of logarithmic functions.
• Logarithmic functions have the form (f(x) = \ln(x)), where (x) is a positive real number.
• The definite integral of a logarithmic function can be evaluated using integration by parts.
• Integration by parts involves choosing one part of the integrand as the “first” part and differentiating it, while integrating the remaining part as the “second” part.

Slide 20

• Example 1: Let’s evaluate the definite integral of (f(x) = \ln(x)) from 1 to 2 using integration by parts. [\int_{1}^{2} f(x)dx = \int_{1}^{2} \ln(x)dx] Let (u = \ln(x)) and (dv = dx). [du = \frac{1}{x}dx \quad \text{and} \quad v = x] Using the integration by parts formula: [\int u dv = uv - \int v du] [\int_{1}^{2} \ln(x)dx = [x\ln(x)]{1}^{2} - \int{1}^{2}x\left(\frac{1}{x}dx\right)] [= \left[2\ln(2) - 1\ln(1)\right] - \left[\int_{1}^{2} dx\right]] [= 2\ln(2) - 1 - (2 - 1)]

Definite integrals of polynomials

• A polynomial is an algebraic expression with variables, coefficients, and exponents.
• The definite integral of a polynomial can be evaluated using the power rule.
• The power rule states: (\int_{a}^{b} ax^n dx = \frac{a}{n+1}(b^{n+1} - a^{n+1})).
• Example 1: (\int_{0}^{2}(2x^3 - 3x^2 + 5x - 2)dx).

Definite integrals of exponential functions

• Exponential functions have the form (f(x) = a^x), where (a) is a constant.
• The definite integral of an exponential function can be evaluated using logarithmic substitution.
• Logarithmic substitution: (u = \ln(a)x) and (du = \frac{\ln(a)}{x}dx).
• Example 1: (\int_{0}^{1}e^xdx).

Definite integrals of trigonometric functions

• Trigonometric functions, like sine, cosine, and tangent, often appear in definite integrals.
• Various trigonometric identities and properties can be used to evaluate definite integrals.
• Example 1: (\int_{0}^{\pi}\sin(x)dx).
• Example 2: (\int_{0}^{\frac{\pi}{2}}\cos(x)dx).

Definite integrals of logarithmic functions

• Logarithmic functions take the form (f(x) = \ln(x)), where (x) is a positive real number.
• The definite integral of a logarithmic function can be evaluated using integration by parts.
• Integration by parts involves differentiating one part and integrating the other.
• Example 1: (\int_{1}^{2}\ln(x)dx).

Summary

• Definite integrals of special functions require specific techniques to evaluate.
• Polynomials can be integrated using the power rule.
• Exponential functions can be integrated using logarithmic substitution.
• Trigonometric functions can be integrated using trigonometric identities.
• Logarithmic functions can be integrated using integration by parts.

Summary (continued)

• Example 1: (\int_{0}^{2}(2x^3 - 3x^2 + 5x - 2)dx).
• Example 2: (\int_{0}^{1}e^xdx).
• Example 3: (\int_{0}^{\pi}\sin(x)dx).
• Example 4: (\int_{0}^{\frac{\pi}{2}}\cos(x)dx).
• Example 5: (\int_{1}^{2}\ln(x)dx).

Tips for solving definite integrals

• Understand the properties and rules of each special function.
• Identify the appropriate technique for the given integral.
• Use correct substitution methods where applicable.
• Pay attention to integration limits.
• Check your answers by differentiating the result.

Tips for solving definite integrals (continued)

• Practice solving various examples and exercises.
• Memorize important identities and formulas.
• Take note of common patterns and shortcuts.
• Work on improving your algebraic manipulation skills.
• Seek help or clarification when needed.

Practice questions

1. Evaluate (\int_{0}^{4}(3x^2 - 2x + 1)dx).

2. Find (\int_{1}^{3}e^xdx).

3. Calculate (\int_{0}^{\frac{\pi}{4}}\sin(x)dx).

4. Determine (\int_{0}^{\frac{\pi}{6}}\cos(x)dx).

5. Solve (\int_{1}^{4}\ln(x)dx).

Practice questions (continued)

1. Evaluate (\int_{0}^{4}(3x^2 - 2x + 1)dx).

2. Find (\int_{1}^{3}e^xdx).

3. Calculate (\int_{0}^{\frac{\pi}{4}}\sin(x)dx).

4. Determine (\int_{0}^{\frac{\pi}{6}}\cos(x)dx).

5. Solve (\int_{1}^{4}\ln(x)dx).