Question
Asked Nov 6, 2019
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Evaluate the integral using the indicated trigonometric substitution
9x3 V9 x2 dx,
x 3 sin(0)
Note: Use an upper-case "C" for the constant of integration.
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Evaluate the integral using the indicated trigonometric substitution 9x3 V9 x2 dx, x 3 sin(0) Note: Use an upper-case "C" for the constant of integration.

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Expert Answer

Step 1

Given,

9x2 9-x d , x=3sin(0)
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9x2 9-x d , x=3sin(0)

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Step 2

On simplificaton

Differentiating x = 3sin (0) with respect to 0, we get
3cos(0) dx = 3cos (0)- de
de
--9x 9-x d=-9[(3sin (0)) 9-(3sin (0)
-3cos(e)de
=-729/sin' (0)/9-9sin2(0) cos (0)de
=-729 sin' (0),/9(1-sin2 (0)) cos (0)de
=-2187sin' (0) cos (0)-cos (0)de
--2187/sin' (e) cos' (0) de
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Differentiating x = 3sin (0) with respect to 0, we get 3cos(0) dx = 3cos (0)- de de --9x 9-x d=-9[(3sin (0)) 9-(3sin (0) -3cos(e)de =-729/sin' (0)/9-9sin2(0) cos (0)de =-729 sin' (0),/9(1-sin2 (0)) cos (0)de =-2187sin' (0) cos (0)-cos (0)de --2187/sin' (e) cos' (0) de

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Step 3

Now

...
=-2187sin' (0) cos (0)- sin(0)de
--2187 (1-cos (0)) cos (0)-sin(0) de
Put cos(e) t-sin(0) d0 = dt sin (0) d0= dt
-2187 (1-cos' (0))cos' (e) sin (0)do= -2187/(1-f).f (-dt)
-2187 (-)dt
=2187
3
+C
5
cos' (0) cos (e)
+C
2187
5
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=-2187sin' (0) cos (0)- sin(0)de --2187 (1-cos (0)) cos (0)-sin(0) de Put cos(e) t-sin(0) d0 = dt sin (0) d0= dt -2187 (1-cos' (0))cos' (e) sin (0)do= -2187/(1-f).f (-dt) -2187 (-)dt =2187 3 +C 5 cos' (0) cos (e) +C 2187 5

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Tagged in

Math

Calculus

Integration