Chapter 5: Integrals
Section 5.3 The Fundamental Theorem of Calculus
Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus:
Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus Part I Quickies d dx 0 x sin (t) d dt t dx Find F (x) if F x = cos u 2 du e x x 4 e t2 dt
Sec. 5.3: The Fundamental Theorem of Calculus Ex 4: Find 7 a) d ln w dw b) d dx dx x 1 x sinh (t) t + t 2 dt c) Find F (x) if lnx F x = e cos (t) dt sinx
Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus:
Sec. 5.3: The Fundamental Theorem of Calculus Ex 5: Find 3 0 a) x e x 2 dx π b) sinx dx 0 c) The area between the curve f x = x 3 2x 2 3x and the x-axis from x = -2 to x = 6. d) 9 1 1 x dx π/6 e) sec x tan (x) dx π/4
Section 5.5 The Substitution Rule (u-substitution)
We know how to find the derivative of any combination of functions Sum rule Difference rule Constant multiple rule Product rule Quotient rule Chain rule
We would like to know how to find the antiderivative of any combination of functions, but only a few rules exist Sum rule Difference rule Constant multiple rule Product rule (no such rule) Quotient rule (no such rule) Chain rule (no such rule) This makes antiderivatives more difficult. We can t find the antiderivative of just any function
What we re going to do is learn integration techniques: develop rules so that if you are in certain very specific situations, you will be able to find the antiderivative The substitution rule (today) Integration by parts (Calc. II) Integration by partial fractions (Calc. II) Trig. substitution (Calc. II)
U-Substitution: The idea is to reverse the chain rule Remember that whenever you know the derivative of a function, you know an antiderivative of another function Ex: Since sin x 2 = 2x cos x 2, we have 2x cos x 2 dx = sin x 2 + C
When to use u-substitution (hints): When it works! If the derivative of part of the function you re integrating is another part of the function you re integrating The derivative of part of the function needs to be multiplied by that part The number in front of the derivative doesn t matter Make u the part of the integrand (thing you re integrating) whose derivative is also a part of the thing you re integrating
What performing a u-substitution: Rewrite the integral so that it only has u s in it Make sure the new integral is simpler than the original integral
U-Substitution To Find An Indefinite Integral: Let u be the appropriate part of the function Find du, then solve for dx Rewrite the integral in terms of u (at the end of this step only u should appear in the integral) The new u integral should be simpler than the original integral Integrate with respect to u At the end, replace u with what you said u equals at the beginning of the problem because the final answer should have x s in it, not u s. Don t forget the +C
U-Substitution To Find A Definite Integral: Start off as if you are calculating an indefinite integral. Then there are 2 ways to go 1) x-world u-world x-world plug in the numbers 2) x-world u-world change the limits of integration to u numbers and never go back to x world
Look at some problems Sec. 5.5: The Substitution Rule 2x sin x 2 dx
Look at some problems Sec. 5.5: The Substitution Rule cos x e sin x dx
Look at some problems Sec. 5.5: The Substitution Rule sin 1 x 1 x 2 dx
Look at some problems Sec. 5.5: The Substitution Rule (ln x) 2 x dx
Look at some problems Sec. 5.5: The Substitution Rule 3x 2 2x 3 + 7 dx
Look at some problems Sec. 5.5: The Substitution Rule 18x 3 3x 2 x + 4 4 dx
Look at some problems x 1 4x 2 dx
Look at some problems Sec. 5.5: The Substitution Rule tan x dx
Look at some problems Sec. 5.5: The Substitution Rule sin 2 x cos x dx
Look at some problems Sec. 5.5: The Substitution Rule sin ( x x) dx
Look at some problems Sec. 5.5: The Substitution Rule sin x sin cos x dx
Look at some problems Sec. 5.5: The Substitution Rule 5 t sin 5 t dt
Look at some problems Sec. 5.5: The Substitution Rule sec 2 x tan 2 x dx
Look at some problems Sec. 5.5: The Substitution Rule sin 2x 1 + cos 2 x dx
Look at some problems e 1/x x 2 dx
Ex 1: Find 2x sin x 2 dx
Ex 2: Find 3x 2 2x 3 + 7 dx 1 1
Ex 3: Find 18x 3 3x 2 x + 4 4 dx
Ex 4: Find Sec. 5.5: The Substitution Rule π 4 π 3 tan x dx
Other Times You Might Use u-substitution Sometimes you make u a part of the thing you are integrating whose derivative is just a number The integral needs to end up simpler than the original integral But!!! Don t ever make u = x because this does nothing!!!
Other Times You Might Use u-substitution 3x 7 dx
Other Times You Might Use u-substitution e 4 9x dx
Ex 5: Find 3x 7 dx
Ex 6: Find 2 1 e 4 9x dx
Other Times You Might Use u-substitution In general, if after making a u substitution you end up with a simpler integral that you know how to integrate, then do it that way Make sure you are able to get rid of all x terms
Other Times You Might Use u-substitution x dx 1 + 2x
Other Times You Might Use u-substitution x 3 x 2 + 1 dx
Ex 7: Find x 1 + 2x dx
Ex 8: Find x 3 x 2 + 1 dx 1 0