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)