Indefinite Integral - Evaluation of integral (Method of substitution)

  • In calculus, the evaluation of indefinite integrals can sometimes be challenging.
  • The method of substitution is a powerful technique to simplify complex integrals.
  • It involves substituting a new variable in place of the original variable.
  • This method helps to transform the integrand into a new function that is easier to integrate.

Steps for solving indefinite integrals using substitution:

  1. Identify a suitable substitution: Choose a new variable that simplifies the integrand.
  1. Compute the derivative: Determine the differential of the new variable.
  1. Rewrite the integral: Express the original integral in terms of the new variable.
  1. Evaluate the integral: Integrate the transformed function with respect to the new variable.
  1. Substitute back: Replace the new variable with the original variable to obtain the final solution.

Example 1:

Evaluate the following indefinite integral: ∫ (2x + 3)^4 dx Solution: Let u = 2x + 3 Now, differentiating both sides with respect to x: du/dx = 2 Rearranging, we have: dx = 1/2 du Substituting these expressions back into the integral: ∫ (2x + 3)^4 dx = ∫ u^4 (1/2 du)

Example 1 (Continued):

Now, we can simplify the integral further: ∫ u^4 (1/2 du) = (1/2) ∫ u^4 du Integrating with respect to u: = (1/2) * (u^5/5) + C Substituting back the value of u: = (1/2) * (2x + 3)^5/5 + C Hence, the solution to the integral is: ∫ (2x + 3)^4 dx = (1/10) * (2x + 3)^5 + C

Example 2:

Evaluate the following indefinite integral: ∫ x^2 √(x^3 + 1) dx Solution: Let u = x^3 + 1 Differentiating both sides with respect to x: du/dx = 3x^2 Rearranging, we have: dx = (1/3x^2) du Substituting these expressions back into the integral: ∫ x^2 √(x^3 + 1) dx = ∫ (x^2) √u (1/3x^2) du

Example 2 (Continued):

Now, we can simplify the integral further: ∫ (x^2) √u (1/3x^2) du = (1/3) ∫ √u du Integrating with respect to u: = (1/3) * (2/3) * u^(3/2) + C Substituting back the value of u: = (1/3) * (2/3) * (x^3 + 1)^(3/2) + C Hence, the solution to the integral is: ∫ x^2 √(x^3 + 1) dx = (2/9) * (x^3 + 1)^(3/2) + C

Key points to remember:

  • The method of substitution simplifies integrals by introducing a new variable.
  • Aim to choose a new variable that makes the integrand simpler.
  • The differential of the new variable should be present in the original integral.
  • After integrating with respect to the new variable, substitute back to the original variable.
  • Always include the constant of integration (C) when evaluating indefinite integrals.

Advantages of using the method of substitution:

  • It can handle complex integrals by transforming them into simpler forms.
  • Makes it easier to identify patterns and solve specific types of integrals.
  • Enables the use of basic integral formulas and rules effectively.
  • Provides a systematic approach to solving integration problems.
  • Helps in determining antiderivatives efficiently.

Limitations of the method of substitution:

  • It may not always yield a simplification of the integral.
  • Choosing an inappropriate substitution can result in a more complicated expression.
  • The method may not work for all types of integrals.
  • Sometimes, additional algebraic manipulation is required before substitution.
  • Careful consideration of the integral and substitution choice is essential for success.

Summary:

  • The method of substitution is a valuable tool for evaluating indefinite integrals.
  • The steps involved are identifying a suitable substitution, rewriting the integral, integrating the transformed function, and substituting back.
  • Examples demonstrated the application of this method and highlighted the importance of proper substitution selection.
  • Advantages and limitations of the method were discussed to provide a comprehensive understanding.

Basic rules and formulas for indefinite integration:

  • Constant rule: ∫ kdx = kx + C, where k is a constant.
  • Power rule: ∫ xn dx = (1/(n+1)) * x^(n+1) + C, where n ≠ -1.
  • Sum rule: ∫ (f(x) + g(x)) dx = ∫ f(x) dx + ∫ g(x) dx.
  • Difference rule: ∫ (f(x) - g(x)) dx = ∫ f(x) dx - ∫ g(x) dx.
  • Constant multiple rule: ∫ k * f(x) dx = k ∫ f(x) dx, where k is a constant.

Common integration formulas:

  • Exponential function: ∫ e^x dx = e^x + C.
  • Logarithmic function: ∫ (1/x) dx = ln|x| + C.
  • Trigonometric function: ∫ sin(x) dx = -cos(x) + C, and ∫ cos(x) dx = sin(x) + C.
  • Inverse trigonometric function: ∫ (1/√(1-x^2)) dx = arcsin(x) + C, and ∫ (1/√(1+x^2)) dx = arctan(x) + C.

Example 3:

Evaluate the following indefinite integral: ∫ (2x + 5) dx Solution: Using the sum rule, we can split the integral: ∫ (2x + 5) dx = ∫ 2x dx + ∫ 5 dx Applying the power rule and constant rule to each part: = (2/2) * x^2 + 5x + C = x^2 + 5x + C Hence, the solution to the integral is: ∫ (2x + 5) dx = x^2 + 5x + C

Example 4:

Evaluate the following indefinite integral: ∫ (4x^3 - 2x^2 + 3x - 1) dx Solution: Using the sum rule, we can split the integral: ∫ (4x^3 - 2x^2 + 3x - 1) dx = ∫ 4x^3 dx - ∫ 2x^2 dx + ∫ 3x dx - ∫ 1 dx Applying the power rule and constant rule to each part: = (4/4) * x^4 - (2/3) * x^3 + (3/2) * x^2 - x + C = x^4 - (2/3) * x^3 + (3/2) * x^2 - x + C Hence, the solution to the integral is: ∫ (4x^3 - 2x^2 + 3x - 1) dx = x^4 - (2/3) * x^3 + (3/2) * x^2 - x + C

Example 5:

Evaluate the following indefinite integral: ∫ (2x^2 - 4x + 7) dx Solution: Using the sum rule, we can split the integral: ∫ (2x^2 - 4x + 7) dx = ∫ 2x^2 dx - ∫ 4x dx + ∫ 7 dx Applying the power rule and constant rule to each part: = (2/3) * x^3 - 2x^2 + 7x + C Hence, the solution to the integral is: ∫ (2x^2 - 4x + 7) dx = (2/3) * x^3 - 2x^2 + 7x + C

Example 6:

Evaluate the following indefinite integral: ∫ (x + 1/x) dx Solution: Using the sum rule, we can split the integral: ∫ (x + 1/x) dx = ∫ x dx + ∫ (1/x) dx Applying the power rule and logarithmic function to each part: = (1/2) * x^2 + ln|x| + C Hence, the solution to the integral is: ∫ (x + 1/x) dx = (1/2) * x^2 + ln|x| + C

Example 7:

Evaluate the following indefinite integral: ∫ (2^x + 1/x) dx Solution: Using the sum rule, we can split the integral: ∫ (2^x + 1/x) dx = ∫ 2^x dx + ∫ (1/x) dx Applying the exponential rule and logarithmic function to each part: = (1/ln(2)) * 2^x + ln|x| + C Hence, the solution to the integral is: ∫ (2^x + 1/x) dx = (1/ln(2)) * 2^x + ln|x| + C

Example 8:

Evaluate the following indefinite integral: ∫ (cos(x) + sin(x)) dx Solution: Using the sum rule, we can split the integral: ∫ (cos(x) + sin(x)) dx = ∫ cos(x) dx + ∫ sin(x) dx Applying the trigonometric function rule to each part: = -sin(x) - cos(x) + C Hence, the solution to the integral is: ∫ (cos(x) + sin(x)) dx = -sin(x) - cos(x) + C

Example 9:

Evaluate the following indefinite integral: ∫ (10e^x - 5ln(x)) dx Solution: Using the sum rule, we can split the integral: ∫ (10e^x - 5ln(x)) dx = ∫ 10e^x dx - ∫ 5ln(x) dx Applying the exponential function and logarithmic function to each part: = 10e^x - 5xln|x| + C Hence, the solution to the integral is: ∫ (10e^x - 5ln(x)) dx = 10e^x - 5xln|x| + C

Summary:

  • In this lecture, we learned various rules and formulas for evaluating indefinite integrals.
  • The basic rules include the constant rule, power rule, sum rule, difference rule, and constant multiple rule.
  • Common integration formulas for exponential, logarithmic, trigonometric, and inverse trigonometric functions were discussed.
  • Several examples demonstrated the application of these rules and formulas.
  • Remember to always include the constant of integration (C) when evaluating indefinite integrals.

Important Concepts in Integration:

  • Integration is the reverse process of differentiation.
  • It involves finding the antiderivative of a function.
  • The indefinite integral symbol is denoted by ∫.
  • The integral of a function represents the area under the curve.

Properties of Integrals:

  • Linearity: ∫ (af(x) + bg(x)) dx = a∫ f(x) dx + b∫ g(x) dx, where a and b are constants.
  • Integration by parts: ∫ u dv = uv - ∫ v du.
  • Integration of a constant: ∫ k dx = kx + C, where k is a constant.
  • Substitution property: If F’(x) = f(x), then ∫ f(g(x))*g’(x) dx = F(g(x)) + C.

Integration Techniques:

  1. Direct substitution: Used to evaluate integrals of the form ∫ f(g(x)) * g’(x) dx.
  1. Integration by parts: Used to evaluate the integral of a product of two functions.
  1. Trigonometric substitution: Used to simplify integrals involving radical expressions.
  1. Partial fractions: Used to decompose a rational function into simpler fractions.
  1. Trigonometric identities: Used to simplify trigonometric integrals.

Example 1: Direct Substitution

Evaluate the following integral: ∫ (3x^2 + 2x + 1) dx Solution: Let u = 3x^2 + 2x + 1 Differentiating both sides: du/dx = 6x + 2 Rearranging and substituting back: dx = (1/(6x + 2)) du Substituting the expressions back into the integral: ∫ (3x^2 + 2x + 1) dx = ∫ u du Integrating with respect to u: = (1/2) * u^2 + C Substituting back the value of u: = (1/2) * (3x^2 + 2x + 1)^2 + C Hence, the solution to the integral is: ∫ (3x^2 + 2x + 1) dx = (1/2) * (3x^2 + 2x + 1)^2 + C

Example 2: Integration by Parts

Evaluate the following integral: ∫ xsin(x) dx Solution: Using the integration by parts formula: ∫ u dv = uv - ∫ v du Let u = x and dv = sin(x) dx Differentiating u and integrating dv: du = dx and v = -cos(x) Applying the formula: ∫ xsin(x) dx = -xcos(x) - ∫ (-cos(x)) dx = -xcos(x) + sin(x) + C Hence, the solution to the integral is: ∫ xsin(x) dx = -xcos(x) + sin(x) + C

Example 3: Trigonometric Substitution

Evaluate the following integral: ∫ (1/√(9 - x^2)) dx Solution: Let x = 3sin(u) Differentiating x and substituting in the integral: dx = 3cos(u) du Substituting back into the integral: ∫ (1/√(9 - x^2)) dx = ∫ (1/√(9 - 9sin^2(u))) * 3cos(u) du = ∫ (3cos(u))/(3cos(u)) du Simplifying the integral: = ∫ du = u + C Substituting back the value of u: = sin^(-1)(x/3) + C Hence, the solution to the integral is: ∫ (1/√(9 - x^2)) dx = sin^(-1)(x/3) + C

Example 4: Partial Fractions

Evaluate the following integral: ∫ (5x + 3)/(x^2 + 4x + 3) dx Solution: The denominator can be factored as (x+1)(x+3). Using partial fraction decomposition: (5x + 3)/(x^2 + 4x + 3) = A/(x + 1) + B/(x + 3) Multiplying through by (x+1)(x+3):

5x + 3 = A(x+3) + B(x+1) Solving for A and B by equating coefficients: A = 1 and B = 4 Substituting back into the integral: ∫ (5x + 3)/(x^2 + 4x + 3) dx = ∫ (1/(x + 1)) dx + ∫ (4/(x + 3)) dx Integrating each term separately: = ln|x + 1| + 4 ln|x + 3| + C Hence, the solution to the integral is: ∫ (5x + 3)/(x^2 + 4x + 3) dx = ln|x + 1| + 4 ln|x + 3| + C

Example 5: Trigonometric Identities

Evaluate the following integral: ∫ cos^2(x) dx Solution: Using the trigonometric identity: cos^2(x) = (1/2) * (1 + cos(2x)) Substituting back into the integral: ∫ cos^2(x) dx = ∫ (1/2) * (1 + cos(2x)) dx Simplifying the integral: = (1/2) * (x + (1/2) * sin(2x)) + C Hence, the solution to the integral is: ∫ cos^2(x) dx = (1/2) * (x + (1/2) * sin(2x)) + C

Recap:

  • Integration involves finding the antiderivative of a function.
  • Properties of integrals include linearity, integration by parts, substitution, and more.
  • Various techniques such as direct substitution, integration by parts, trigonometric substitution, partial fractions, and trigonometric identities can simplify integrals.
  • Examples were provided to illustrate the application of each technique.
  • By using these techniques, we can solve a wide range of integration problems.