Integrate 2sin(x)cos(x)dx
Let u = cos(x) and du = -sin(x)dx and pull out the -2:
-2[Integral(u*du)]
Integrate with respect to u:
-2(u2)/2 + C
Simplify:
-u2 + C
Replace u with cos(x):
-cos2(x) + C
-4
2sinxcosx-cosx=0 Factored : cosx(2sinx-1)=0 2 solutions: cosx=0 or sinx=.5 For cosx=0, x=90 or 270 degrees For sinx=.5, x=30 degrees x = {30, 90, 270}
1/3ln(sin3x) + C
.2x^5+x+C
e 2x = (1/2) e 2x + C ============
-4
sin(2x) = 2sinxcosx
d/dx (sin x)^2=2sinxcosx
1
Start by squaring both sides of the equation to get, (sinx + cosx)2 = 0.25 Simplify the left side to get sin2x + 2sinxcosx + cos2x = 0.25 Using the Pythagorean identity gets 2sinxcosx + 1 = 0.25 2sinxcosx = -0.75 Using the double angle formula gets Sin(2x) = -0.75 Take the arcsin to get 2x = sin-1(-0.75) x = sin-1(-0.75)/2 Now, a scientific calculator can be used to find the solutions.
2sinxcosx-cosx=0 Factored : cosx(2sinx-1)=0 2 solutions: cosx=0 or sinx=.5 For cosx=0, x=90 or 270 degrees For sinx=.5, x=30 degrees x = {30, 90, 270}
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to integrate = sheelev (שילב)
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