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| Integral
calculus |
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The
indefinite integral |
Trigonometric
integrals
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Trigonometric
integrals of the form
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∫
sin (m x) sin (n x) dx,
∫
sin
(m x)
cos (n x)
dx,
∫
cos (m x)
cos (n x)
dx.
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Trigonometric
integrals of the form
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∫
sin (m x) sin (n x) dx,
∫
sin
(m x)
cos (n x)
dx,
∫
cos(m x)
cos (n x)
dx,
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in these cases we use the following product to sum formulas,
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and the addition formulas.
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We
can use the above formula 3) or derive it from the addition
formulas, thus
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Integrals
of rational functions containing sine and cosine, ∫
R
(sin x, cos x) dx
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Integrals
of the form
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∫
R
(sin x, cos x)
dx
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where
R
denotes a rational function of sin
x
and cos
x,
can be transformed to a rational function of the new variable t,
using substitution tan
(x/2) = t. Then,
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therefore,
by substituting
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In
case the integrand expression does not change the sign when both
the sine and the cosine functions change
the sign, i.e., if
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R
(-
sin x, -
cos x)
or
R
(sin x, cos x)
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we
can use the substitution tan
x = t. Therefore,
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| Calculus contents
F |
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