Indefinite Integral Practice - Edlˆ’2cos 3sinxxC++ b. ttC++csc c. −−cot sinθθC tC+ d. 3 cos...
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Indefinite Integral Practice Name: 1. Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. a. 4 3 9 3 dx C x x ⎛ ⎞ − = + ⎜ ⎟ ⎝ ⎠ ∫ b. 3 2 3 1 1 3 dx x x C x ⎛ ⎞ − = − + ⎜ ⎟ ⎝ ⎠ ∫ 2. Integrate. a. 6dx ∫ b. 2 3 t dt ∫ c. 3 5 x dx − ∫ d. du ∫ e. 3 2 x dx ∫ f. 3 xdx ∫ g. 1 dx x x ∫ h. 3 1 2 dx x ∫ i. ( ) 3 2 x dx + ∫ j. ( ) 4 3 2 3 1 x x dx + − ∫ k. 2 3 x dx ∫ l. 3 1 dx x ∫ m. 2 1 4 dx x ∫ n. 2 2 2 t dt t + ∫ o. ( ) 2 3 1 u u du + ∫ p. ( )( ) 1 6 5 x x dx − − ∫ q. 2 y ydy ∫ 3. Given below are the derivative graph, f x () , find a possible equation for Fx () the antiderivative of f x () a. b. Answers: (Of course, you could have checked all of yours using differentiation!) 2. a. 6 x C + b. 3 t C + c. 2 5 2 C x − + d. u C + e. 5 2 2 5 x C + f. 4 3 3 4 x C + g. 2 C x − + h. 2 1 4 C x − + i. 4 2 4 x x C + + j. 7 2 3 6 3 7 2 x x x C + − + k. 5 3 3 5 x C + l. 2 1 2 C x − + m. 1 4 C x − + n. 2 t C t − + o. 4 2 3 1 4 2 u u C + + p. 3 2 11 2 5 2 x x x C − + + q. 7 2 2 7 y C + 3. Example solution a. Fx () = 2 x + 3 b. Fx () = 1 2 x 2 + 5
Transcript of Indefinite Integral Practice - Edlˆ’2cos 3sinxxC++ b. ttC++csc c. −−cot sinθθC tC+ d. 3 cos...
Indefinite Integral Practice Name: 1. Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side.
a. 4 3
9 3dx Cx x
⎛ ⎞− = +⎜ ⎟⎝ ⎠∫ b. 323
11 3dx x x Cx
⎛ ⎞− = − +⎜ ⎟
⎝ ⎠∫
2. Integrate.
a. 6dx∫ b. 23t dt∫ c. 35x dx−∫ d. du∫
e. 32x dx∫ f. 3 xdx∫ g. 1 dx
x x∫ h. 3
12
dxx∫
i. ( )3 2x dx+∫ j. ( )432 3 1x x dx+ −∫ k. 23 x dx∫ l. 3
1 dxx∫
m. 2
14
dxx∫ n.
2
2
2t dtt+
∫ o. ( )23 1u u du+∫
p. ( )( )1 6 5x x dx− −∫ q. 2y ydy∫
3. Given below are the derivative graph, f x( ) , find a possible equation for F x( ) the antiderivative of f x( )
a.
b.
Answers: (Of course, you could have checked all of yours using differentiation!)
2. a. 6x C+ b. 3t C+ c. 2
52
Cx
− + d. u C+
e. 522
5x C+ f. 433
4x C+ g. 2 C
x− + h. 2
14
Cx
− +
i. 4
24x x C+ + j.
7 236 37 2x x x C+ − + k.
533
5x C+ l. 2
12
Cx
− + m. 14
Cx
− +
n. 2t Ct
− + o. 4 23 14 2u u C+ +
p. 3 2112 52
x x x C− + + q. 722
7y C+
3. Example solution a. F x( ) = 2x + 3 b. F x( ) = 12x2 + 5
Initial Condition & Integration of Trig Functions Practice
1. Find the particular solution ( )y f x= that satisfies the differential equation and initial condition.
a. '( ) 3 3, (1) 4f x x f= + = b. '( ) 6 ( 1), (10) 10f x x x f= − = −
c. 3
2 3'( ) , 0, (2)4
xf x x fx−= > = d. 2'( ) sec , 2 3
3f x x f π⎛ ⎞= =⎜ ⎟⎝ ⎠
2. Find the equation of the function f whose graph passes through the given point.
a. ( )'( ) 6 10, 4,2f x x= − b. f '(x) = 3x2 + e2x − 1
ex , f 0( ) = 3
3. Find the function f that satisfies the given conditions.
a. "( ) 2, '(2) 5, (2) 10f x f f= = = b. f "(x) = x−2 3, f '(8) = 6, f (0) = 0 4. Integrate.
a. (2sin 3cos )x x dx+∫ b. ( )1 csc cott t dt−∫
c. ( )2csc cos dθ θ θ−∫ d. ( )2 sint t dt−∫
5.
Answers:
1. a. 32( ) 2 3 1f x x x= + − b. 3 2( ) 2 3 1710f x x x= − −
c. 2
1 1 1( )2
f xx x−= + + d. ( ) tan 3f x x= +
2. a. 32( ) 4 10 10f x x x= − + b.
f (x) = x3 + 1
2e2x + e− x +1.5
3. a. 2( ) 4f x x x= + + b. 439( )
4f x x=
4. a. 2cos 3sinx x C− + + b. csct t C+ +
c. cot sin Cθ θ− − + d. 3
cos3t t C+ +
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