For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 107. ∫ 0 2 π sin x sin ( 2 x ) sin ( 3 x ) d x
For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 107. ∫ 0 2 π sin x sin ( 2 x ) sin ( 3 x ) d x
For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible.
107.
∫
0
2
π
sin
x
sin
(
2
x
)
sin
(
3
x
)
d
x
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Use the substitution rule for definite integrals to evaluate the following definite integral.
J
cos x sinx dx
/2
3
cos xsin x dx
(Simplify your answer.)
/2
dx
x² +9
2.
Use an appropriate trigonometric substitution to rewrite J
as
J(a simplified trig function of 0) d0. Carefully show your work
with differentials and use of trig identities to help simplify.
Then find the integral. Show any use of substitutions and reasoning with right triangles.
Evaluate the following definite and indefinite integrals (if they exist). Show all of your work for full credit:
[(-8x° +9x² – 12.x)dx
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Fundamental Theorem of Calculus 1 | Geometric Idea + Chain Rule Example; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=hAfpl8jLFOs;License: Standard YouTube License, CC-BY