Indefinite Integral - BASIC INTRODUCTION ON INTEGRALS

• In calculus, an integral is an antiderivative, which represents the reverse of the process of differentiation.

• An indefinite integral of a function f(x) is denoted as ∫f(x)dx.

• It represents a family of functions that differ by a constant.

• The integral of a function gives us the area under the curve between the curve and the x-axis.

• This process is also known as finding the area of the region bounded by the curve.

• It can be thought of as the summation of infinitely small rectangles under the curve.

• The indefinite integral of a function f(x) is computed using integral formulas and mathematical techniques.

• Different functions have different integral formulas.

• Some common integral formulas include power rule, trigonometric integrals, and exponential integrals.

• The integral of a constant is the constant multiplied by the variable: ∫kdx = kx + C, where k is a constant and C is the constant of integration.

• For example, ∫5dx = 5x + C.

• The integral of x to the power of n is obtained by increasing the power by 1 and dividing by the new power: ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1 and C is the constant of integration.

• For example, ∫x^2 dx = (x^3)/3 + C.

• The integral of a sum or difference of functions is the sum or difference of their integrals: ∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx.

• For example, ∫(3x^2 + 4x)dx = ∫3x^2 dx + ∫4x dx.

• The integral of a constant multiplied by a function is equal to the constant multiplied by the integral of the function: ∫k f(x) dx = k ∫f(x)dx, where k is a constant.

• For example, ∫5x^2 dx = 5 ∫x^2 dx.

• The integral of a constant multiplied by the variable to the power of n is obtained by increasing the power by 1, dividing by the new power, and multiplying by the constant: ∫k x^n dx = k(x^(n+1))/(n+1) + C, where n ≠ -1 and C is the constant of integration.

• For example, ∫4x^3 dx = 4(x^4)/4 + C.

• The integral of a constant multiplied by a function raised to the power of n is obtained by dividing the function by the new power, multiplying by the constant, and adding 1 to the power: ∫k [f(x)]^n dx = k ∫[f(x)]^(n+1)/(n+1) + C, where n ≠ -1 and C is the constant of integration.

• For example, ∫8(x^2)³ dx = 8 ∫(x^2)^4/4 + C.

• The integral of the sum or difference of two functions multiplied by each other is not as straightforward as the previous cases.

• It involves applying the technique of integration by parts, which is beyond the scope of this basic introduction.

• Integration by parts involves integrating one function while differentiating the other.

11. Basic Integration Rules

• The integral of a constant is the constant multiplied by the variable: ∫kdx = kx + C, where k is a constant and C is the constant of integration.
• Example: ∫4dx = 4x + C.

12. Basic Integration Rules

• The integral of x to the power of n is obtained by increasing the power by 1 and dividing by the new power: ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1 and C is the constant of integration.
• Example: ∫x^3 dx = (x^4)/4 + C.

13. Basic Integration Rules

• The integral of a sum or difference of functions is the sum or difference of their integrals: ∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx.
• Example: ∫(3x^2 + 4x)dx = ∫3x^2 dx + ∫4x dx.

14. Basic Integration Rules

• The integral of a constant multiplied by a function is equal to the constant multiplied by the integral of the function: ∫k f(x) dx = k ∫f(x)dx, where k is a constant.
• Example: ∫5x^2 dx = 5 ∫x^2 dx.

15. Basic Integration Rules

• The integral of a constant multiplied by the variable to the power of n is obtained by increasing the power by 1, dividing by the new power, and multiplying by the constant: ∫k x^n dx = k(x^(n+1))/(n+1) + C, where n ≠ -1 and C is the constant of integration.
• Example: ∫4x^3 dx = 4(x^4)/4 + C.

16. Basic Integration Rules - Examples

• Example: Find ∫(3x^2 + 5x - 2)dx.
  • Using the basic integration rules, we can write this as ∫3x^2 dx + ∫5x dx - ∫2 dx.
  • Applying the power rule, we get (x^3)/3 + (5x^2)/2 - 2x + C, where C is the constant of integration.

17. Basic Integration Rules - Examples

• Example: Find ∫(4sin(x) + 3cos(x))dx.
  • Using the basic integration rules, we can write this as ∫4sin(x) dx + ∫3cos(x) dx.
  • The integral of sin(x) is -cos(x), and the integral of cos(x) is sin(x).
  • So, the final result becomes -4cos(x) + 3sin(x) + C, where C is the constant of integration.

18. Basic Integration Rules - Examples

• Example: Find ∫(e^x + ln(x))dx.
  • Using the basic integration rules, we can write this as ∫e^x dx + ∫ln(x) dx.
  • The integral of e^x is e^x, and the integral of ln(x) is xln(x) - x.
  • So, the final result becomes e^x + xln(x) - x + C, where C is the constant of integration.

19. Basic Integration Rules - Examples

• Example: Find ∫(4x^3 + 2x^2)dx.
  • Using the basic integration rules, we can write this as ∫4x^3 dx + ∫2x^2 dx.
  • Applying the power rule, we get (x^4) + (2x^3)/3 + C, where C is the constant of integration.

20. Basic Integration Rules - Examples

• Example: Find ∫(2x^3 + 3√x)dx.
  • Using the basic integration rules, we can write this as ∫2x^3 dx + ∫3√x dx.
  • Applying the power rule, we get (x^4)/2 + (2x^(5/2))/(5/2) + C.
  • Simplifying further, we have (x^4)/2 + (4x^(5/2))/5 + C, where C is the constant of integration.

21. Basic Integration Rules - Examples

• Example: Find ∫(2sin(x) + 3cos(x))dx.
  • Using the basic integration rules, we can write this as ∫2sin(x)dx + ∫3cos(x)dx.
  • The integral of sin(x) is -cos(x), and the integral of cos(x) is sin(x).
  • So, the final result becomes -2cos(x) + 3sin(x) + C, where C is the constant of integration.

22. Basic Integration Rules - Examples

• Example: Find ∫(6x + 2/x)dx.
  • Using the basic integration rules, we can write this as ∫6x dx + ∫2/x dx.
  • Applying the power rule, we get (3x^2) + 2ln|x| + C, where C is the constant of integration.

23. Basic Integration Rules - Examples

• Example: Find ∫(x^2 + √x + 1)dx.
  • Using the basic integration rules, we can write this as ∫x^2 dx + ∫√x dx + ∫1 dx.
  • Applying the power rule, we get (x^3)/3 + (2x^(3/2))/(3/2) + x + C.
  • Simplifying further, we have (x^3)/3 + (4x^(3/2))/3 + x + C, where C is the constant of integration.

24. Basic Integration Rules - Examples

• Example: Find ∫(4e^x + 2x^2 + 1/x)dx.
  • Using the basic integration rules, we can write this as ∫4e^x dx + ∫2x^2 dx + ∫1/x dx.
  • The integral of e^x is e^x, and applying the power rule to x^2, we get (2x^3)/3.
  • Applying the natural logarithm rule to 1/x, we get ln|x|.
  • So, the final result becomes 4e^x + (2x^3)/3 + ln|x| + C, where C is the constant of integration.

25. Basic Integration Rules - Examples

• Example: Find ∫(3x^2 + x^(1/2) + ln(x))dx.
  • Using the basic integration rules, we can write this as ∫3x^2 dx + ∫x^(1/2) dx + ∫ln(x) dx.
  • Applying the power rule to x^2, we get (x^3)/3.
  • Applying the power rule to x^(1/2), we get (2x^(3/2))/(3/2).
  • Applying the natural logarithm rule to ln(x), we get xln|x|.
  • So, the final result becomes (x^3)/3 + (4x^(3/2))/3 + xln|x| + C, where C is the constant of integration.

26. Basic Integration Rules - Examples

• Example: Find ∫(4x^-1 - 5x^-2)dx.
  • Using the basic integration rules, we can write this as ∫4x^-1 dx - ∫5x^-2 dx.
  • Applying the natural logarithm rule to x^-1, we get 4ln|x|.
  • Applying the power rule to x^-2, we get -5(x^-1)/(-1) = 5x^-1.
  • So, the final result becomes 4ln|x| + 5x^-1 + C, where C is the constant of integration.

27. Basic Integration Rules - Examples

• Example: Find ∫(sin(x)^2 + cos(x)^2)dx.
  • Using the basic integration rules, we can write this as ∫sin(x)^2 dx + ∫cos(x)^2 dx.
  • The integral of sin(x)^2 is (1/2)(x - sin(x)cos(x)).
  • The integral of cos(x)^2 is (1/2)(x + sin(x)cos(x)).
  • So, the final result becomes (1/2)(x - sin(x)cos(x)) + (1/2)(x + sin(x)cos(x)) + C, where C is the constant of integration.

28. Basic Integration Rules - Examples

• Example: Find ∫(4x^3 + 3sin(x))dx.
  • Using the basic integration rules, we can write this as ∫4x^3 dx + ∫3sin(x) dx.
  • Applying the power rule to x^3, we get (x^4)/4.
  • The integral of sin(x) is -cos(x).
  • So, the final result becomes (x^4)/4 - 3cos(x) + C, where C is the constant of integration.

29. Basic Integration Rules - Examples

• Example: Find ∫(2e^x - 3x^2 + 1/x^2)dx.
  • Using the basic integration rules, we can write this as ∫2e^x dx - ∫3x^2 dx + ∫1/x^2 dx.
  • The integral of e^x is e^x, and applying the power rule to x^2, we get -(x^3)/3.
  • Applying the power rule to x^-2, we get -(1/x).
  • So, the final result becomes 2e^x - (x^3)/3 - (1/x) + C, where C is the constant of integration.

30. Basic Integration Rules - Examples

• Example: Find ∫(5x^4 - 2e^x - 1/x)dx.
  • Using the basic integration rules, we can write this as ∫5x^4 dx - ∫2e^x dx - ∫1/x dx.
  • Applying the power rule to x^4, we get (x^5)/5.
  • The integral of e^x is e^x, and the natural logarithm rule to 1/x gives -ln|x|.
  • So, the final result becomes (x^5)/5 - 2e^x - ln|x| + C, where C is the constant of integration.