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QUESTION: 1

The value of is:

Solution:

∫(-3 to 3) (x+1)dx

= ∫(-3 to -1) (x+1)dx + ∫(-1 to 3) (x+1) dx

= [x^{2} + x](-3 to -1) + [x^{2} + x](-1 to 3)

= [½ - 1 - (9/2 - 3)] + [9/2 + 3 - (½ - 1)]

= -[-4 + 2] + [4 + 4]

= -[-2] + [8]

= 10

QUESTION: 2

Solution:

Option d is correct, because it is the property of definite integral

∫02a f(x) dx = ∫0a f(x) dx + ∫0a f(2a – x) dx

QUESTION: 3

Solution:

∫02π|cos x|

[sin x]02π

[sin0 - sin2π]

0 - 0 = 0

QUESTION: 4

Find the value of the integral.

Solution:

QUESTION: 5

The value of is:

Solution:

I = ∫(1+cos^{2}x)/2 dx

⇒ I = 1/2∫(1+cos^{2}x)dx

⇒ I = 1/2∫1dx + 1/2∫cos^{2}xdx

Using the integral formula ∫coskxdx = sinkx/k + c, we have

∫cos^{2} xdx = 1/2x + 1/2(sin^{2}x)/2 + c

⇒ [1/2x + 1/4sin^{2}x](-π/4 to π/4)

=> [-π/8 - π/8] + [1/4sin(-π/2) - 1/4sin(π/2)]

= - 2π/8 + [-¼ - ¼]

= -π/4 -½

= -(π/4 +½)

QUESTION: 6

Solution:

QUESTION: 7

The value of is:

Solution:

QUESTION: 8

Solution:

QUESTION: 9

If f(2a – x) = f(x), then

Solution:

QUESTION: 10

Solution:

For sin^{2}(X), we will use the cos double angle formula:

cos(2X) = 1 - 2sin^{2}(X)

The above formula can be rearranged to make sin^{2}(X) the subject:

sin^{2}(X) = 1/2(1 - cos(2X))

You can now rewrite the integration:

∫sin^{2}(X)dX = ∫1/2(1 - cos(2X))dX

Because 1/2 is a constant, we can remove it from the integration to make the calculation simpler. We are now integrating:

1/2 x ∫(1 - cos(2X)) dX

= 1/2 x (X - 1/2sin(2X)) + C]-pi/4 to pi/4

∫sin2(X) dX = [1/2X - 1/4sin(2X)]-pi/4 to pi/4 + C

½[-pi/2] - 1/4sin(2(-pi/4)] - ½[pi/2] - 1/4sin(2(pi/4)]

= π/2

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