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Lecture Comments (3)

2 answers

Last reply by: Yong Su An
Tue Feb 24, 2015 2:03 PM

Post by Orsolya Krispán on April 10, 2013

you turned the negative sign into a positive one. in example 5.

Integrals Basic Rules of Integration

  • Constant rule:
  • Constant multiple rule:
  • Addition and difference rule:
  • Exact same rules as derivatives!

Integrals Basic Rules of Integration

∫7  dx
  • ∫7  dx = 7 ∫dx
  • ∫7  dx = 7(x + C)
∫7  dx = 7x + c
∫9x  dx
  • ∫9x  dx = 9 ∫x  dx
  • ∫9x  dx = 9([(x2)/2] + C)
∫9x  dx = [9/2] x2 + c
∫(9x + 7) dx
  • ∫(9x + 7) dx = ∫9x  dx + ∫7  dx
  • ∫(9x + 7) dx = 9 ∫x  dx + ∫7  dx
  • ∫(9x + 7) dx = 9 ([(x2)/2] + c1) + 7x + c2
  • ∫(9x + 7) dx = [9/2] x2 + 7x + c3 + c2
  • We can combine the two constants into a new constant
∫(9x + 7) dx = [9/2]x2 + 7x + c
∫(9x − 7) dx
  • ∫9x − 7 dx = ∫9x  dx − ∫7  dx
  • ∫9x − 7 dx = 9 ∫x  dx − ∫7  dx
  • ∫9x − 7 dx = 9 ([(x2)/2] + c1) − 7x − c2
∫9x − 7 dx = [9/2]x2 − 7x + c
∫2 x3 dx + 2 ∫x3 dx
  • ∫2 x3 dx + 2 ∫x3 dx = 2 ∫x3 dx + 2 ∫x3 dx
  • ∫2 x3 dx + 2 ∫x3 dx = 4 ∫x3 dx
  • ∫2 x3 dx + 2 ∫x3 dx = 4 ([(x4)/4] + C)
∫2 x3 dx + 2 ∫x3 dx = x4 + c
∫x5 − x7 dx
  • ∫x5 − x7 dx = ∫x5 dx − ∫x7 dx
  • ∫x5 − x7 dx = [(x6)/6] + c1 − ([(x7)/7] + c2)
∫x5 − x7 dx = [(x6)/6] − [(x7)/7] + c
∫3 x2 − 2x + 5 dx
  • ∫3 x2 − 2x + 5 dx = 3 ∫x2 dx − 2 ∫x  dx + 5 ∫ dx
  • We can skip a couple of steps by combining the constants ahead of time.
  • ∫3 x2 − 2x + 5 dx = 3 [(x3)/3] − 2 [(x2)/2] + 5x + c
∫3 x2 − 2x + 5 dx = x3 − x2 + 5x + c
∫√{4x} dx
  • ∫√{4x} dx = ∫√4 √x dx
  • ∫√{4x} dx = ∫2 √x dx
  • ∫√{4x} dx = 2 ∫x[1/2] dx
  • ∫√{4x} dx = 2 ([(x[1/2] + 1)/([1/2] + 1)]) + c
  • ∫√{4x} dx = 2 ([(x[3/2])/([3/2])]) + c
∫√{4x} dx = [4/3] x[3/2] + c
∫1 − [(x2)/2] + [(x4)/24] dx
  • ∫1 − [(x2)/2] + [(x4)/24] dx = ∫dx − [1/2] ∫x2 dx + [1/24] ∫x4 dx
  • ∫1 − [(x2)/2] + [(x4)/24] dx = x − [1/2] [(x3)/3] + [1/24] [(x5)/5] + c
∫1 − [(x2)/2] + [(x4)/24] dx = x − [(x3)/6] + [(x5)/120] + c
Find [d/dx] (x − [(x3)/6] + [(x5)/120])
  • [d/dx] (x − [(x3)/6] + [(x5)/120]) = 1 − [(3x2)/6] + [(5x4)/120]
[d/dx] (x − [(x3)/6] + [(x5)/120]) = 1 − [(x2)/2] + [(x4)/24]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Integrals Basic Rules of Integration

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Basic Rules of Integration 0:09
    • Constant Rule
    • Example 1
  • Addition and Difference Rule 1:40
    • Example 2
  • Example 3: Subtraction/ Difference Rule 2:47
  • Example 4 3:55
  • Example 5 5:19
  • Example 6 7:37