Learning Outcomes

  1. Find the general antiderivative of a given function.
  2. Explain the terms and notation used for an indefinite integral.
  3. Use anti differentiation to solve simple initial-value problems.
     

Textbook

  • Chapter 4.10 ¬†Antiderivatives ¬†

Textbook Assignment

  • p. 497: ¬† ¬†465, 468-471 all, 473, 476, 477, 481, 482, 490-493 all, 499, 500, 502

WeBWork Assignment

  • Application-Antiderivatives

Exit problems of the session 

  1. Evaluate following indefinite integrals:

    (a).  \displaystyle\int (3e^x-\sin x)dx          (b).   \displaystyle\int \dfrac{2x^6+3x^2}{x^3}dx           (c).   \displaystyle\int x^2(1-2x^3)dx           (d).   \displaystyle\int \sqrt[3]{x^2}dx
  2. Solve the initial value problem:  f'(x)=\sqrt{x}+x^2,   f(0)=2.

 

Key Concepts

  • If ¬†F'(x)=f(x) ¬†for all x in the domain of f, ¬†then ¬†F(x) ¬†is an antiderivative of f(x).
  • The most general antiderivative of f(x) is the indefinite integral of f(x), denoted by the notation \displaystyle\int f(x)dx. ¬†If F(x) ¬†is an antiderivative of f(x), ¬†then¬†

\displaystyle\int f(x) dx=F(x) + C,   C  is any constant

  • Properties of indefinite integrals: Let F(x) and G(x) be antiderivatives of f(x) and g(x), respectively and let k be any real number, then¬†

\displaystyle\int f(x)\pm g(x) dx=F(x) \pm G(x) + C

\displaystyle\int k f(x) dx=kF(x) + C

  • Basic indefinite integral formulas:

             \displaystyle\int k dx=kx + C;                                 \displaystyle\int x^n dx=\dfrac{x^{n+1}}{n+1} + C    n\neq -1

             \displaystyle\int \dfrac{1}{x} dx=\ln|x| + C;                            \displaystyle\int e^x dx=e^x + C

             \displaystyle\int \cos x dx=\sin x + C;                       \displaystyle\int \sin x dx=-\cos  x + C        

             \displaystyle\int \sec^2 x dx=\tan x + C;                    \displaystyle\int \csc^2 x dx=-\cot x + C;

             \displaystyle\int \sec x\tan x dx=\sec x + C;             \int \csc x\cot x dx=-\csc x + C;

             \displaystyle\int \dfrac{1}{\sqrt{1-x^2}} dx=\sin ^{-1}x + C;           \displaystyle\int \dfrac{1}{1+x^2} dx=\tan ^{-1}x + C;  

  • Solve the initial-value problem

\dfrac{dy}{dx}=f(x),    y(x_0)=y_0

requires us first find the set of antiderivatives of f(x) and then to look for the particular antiderivative that also satisfies the initial condition, that is to use initial condition to find integral constant C. 

 

Videos and Practice Problems of Selected Topics

  1. \rhd Antiderivatives and indefinite integrals (3:42) What is the antiderivative of 2x?
  2. * Practice: antiderivatives and indefinite integrals (4 problems)
  3. \rhd Reverse power rule (5:47) Deriving \int x^n dx followed by finding \displaystyle\int x^5 dx and \displaystyle\int 5x^{-2}dx.
  4. * Practice: Reverse power rule. (4 problems)
  5. * Practice: Reverse power rule with negative and fractional powers. (4 problems)
  6. \rhd Indefinite integrals: sums and multiples (4:55) The indefinite integral of a sum and a multiple of a function. Find \displaystyle\int (x^2+\cos x)dx and \displaystyle\int(\pi\cos x)dx.
  7. * Practice: Reverse power rule with sums and multiples. (4 problems)
  8. \rhd Rewriting before integration (5:16) Find \displaystyle\int x^2(3x-1)dx, \displaystyle\int \dfrac{x^3+3x^2-5}{x^2}dx, \displaystyle\int \sqrt[3]x^5dx.
  9. * Practice: Reverse power rule: rewriting before integrating. (4 problems)
  10. \rhd Indefinite integrals of \sin(x), \cos(x) and e^x (4:03) Find \displaystyle\int (\sin t + \cos t)dt and \displaystyle\int \left( e^a+\dfrac{1}{a}\right) da.
  11. * Practice: Indefinite integrals: e^x and \dfrac{1}{x}. (4 problems)
  12. * Practice: Indefinite integrals: \sin x and \cos x. (4 problems)