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Calculus Concepts |
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Basic Integration |
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The
integration of sin(u) du = -cos(u) + C Example: The integration of sin(3x) dx u = 3x, so du = 3 dx Add this 3 by changing the formula to (1/3) integration of sin(3x) 3 dx Therefore the answer is -(1/3) cos(3x) + C |
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The integration of cos(u) du = sin(u) + C Example: The integration of cos(x^2) x dx u = x^2, so du = 2x dx Add this 2 by changing the formula to (1/2) integration of cos(x^2) 2x dx Therefore the answer is (1/2) sin(x^2) + C |
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The
integration of tan(u) du = ln abs[sec(u)] + C Example: The integration of tan(7x) dx u = 7x, so du = 7
dx Add this 7 by
changing the formula to (1/7) integration of tan(7x) 7 dx Therefore the
answer is (1/7) ln abs[sec(7x)] + C |
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The integration
of cot(u) du =
ln abs[sin(u)] + C Example: The integration of cot(2x^2) x dx u = 2x^2, so du = 4x dx Add this 4 by changing the formula to (1/4) integration of cot(2x^2) 4x dx Therefore the answer is (1/4)
ln abs[sin(2x^2)] + C |
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The integration of sec(u) du = ln abs[sec(u) + tan(u)]
+ C Example: The integration of sec(x) dx u = x, so du = dx Nothing needs to be
added in this case so The answer is ln abs[sec(x) + tan(x)] + C |
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The integration of csc(u) du = ln abs[csc(u) –
cot(u)] + C Example: The integration of csc(8x) dx u = 8x, so du = 8
dx Add this 8 by
changing the formula to (1/8) integration of csc(8x) 8 dx Therefore the
answer is (1/8) ln abs[csc(8x) – cot(8x)] + C |
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The integration of [sec(u)]^2 du = tan(u) + C Example: The integration of [sec(3x)]^2 dx u
= 3x, so du = 3 dx Add this 3 by changing the
formula to (1/3) integration of [sec(3x)]^2
3 dx Therefore the answer is (1/3) tan(3x) + C |
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The integration of [sec(u)][tan(u)]
du = sec(u) + C Example: The integration of [sec(10x)][tan(10x)]
dx u = 10x, so du = 10
dx Add this 10 by
changing the formula to (1/10) integration
of [sec(10x)][tan(10x)] 10 dx Therefore the
answer is (1/10) sec(10x) + C |
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The
integration of [csc(u)]^2 du = -cot(u) + C Example: The integration of [csc(x)]^2 dx u = x, so du = dx Nothing needs to be added in this case so The answer is -cot(x) + C |
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The
integration of [csc(u)][cot(u)] du = -csc(u) + C Example: The integration of [csc(9x)][cot(9x)] dx u = 9x, so du = 9 dx Add this 9 by changing the formula to (1/9) integration of [csc(9x)][cot(9x)] 9 dx Therefore the answer is -(1/9) csc(9x) + C |
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The
integration of (1/u) du = ln abs[u] + C Example: The integration of [1/(6x)] dx u
= 6x, so du = 6 dx Add this 6 by
changing the formula to (1/6) integration of [1/(6x)] 6 dx Therefore the
answer is (1/6) ln abs[6x] + C |
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The
integration of e^u du = e ^u + C Example: The integration of e^(5x) dx u
= 5x, so du = 5 dx Add this 5 by
changing the formula to (1/5) integration of e^(5x) 5 dx Therefore the
answer is (1/5) e^(5x) + C |
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Advanced
Integration |
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The integration of arcsin u du = u
arcsin(u) + (1/a) root(1 - u^2) + C |
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The
integration of arctan u du = u
arctan(u) – (1/2) ln(1 + u^2) + C |
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The
integration of [ dx / (x^2 + a^2)] = (1/a)
arctan(x/a) + C |
The integration
of [dx / root(a^2 –
x^2)] = arcsin(x / a) + C |
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Integration |
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