Integration of Circular Functions (VCE SSCE Mathematical Methods): Quizzes

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Integration of circular functions
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What is sin(kx+a)dx\int \sin(kx + a) \, dx equal to?

1kcos(kx+a)+c-\frac{1}{k} \cos(kx + a) + c

What is cos(kx+a)dx\int \cos(kx + a) \, dx equal to?

1ksin(kx+a)+c\frac{1}{k} \sin(kx + a) + c

What does the coefficient kk in sin(kx+a)\sin(kx + a) represent?

Frequency

What does the constant aa in sin(kx+a)\sin(kx + a) represent?

Phase shift (horizontal translation)

Find an antiderivative of sin(3x+π4)\sin(3x + \frac{\pi}{4}).

13cos(3x+π4)+c-\frac{1}{3} \cos(3x + \frac{\pi}{4}) + c

Find an antiderivative of 14sin(4x)\frac{1}{4} \sin(4x).

116cos(4x)+c-\frac{1}{16} \cos(4x) + c

What is the exact value of 0π4sin(2x)dx\int_{0}^{\frac{\pi}{4}} \sin(2x) \, dx?

12\frac{1}{2}

What is the area under y=sin(12x)y = \sin(\frac{1}{2}x) from x=0x = 0 to x=2πx = 2\pi?

44 square units

Why must we divide by kk when integrating sin(kx+a)\sin(kx + a)?

Because of the chain rule working in reverse

How should regions below the x-axis be treated when finding areas?

Take the negative of the integral

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