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S(sin^4(x)cos^3(x) Trigonometric Integrals

S(sin4(x)cos3(x)   S=INTEGRAL

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Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
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∫(sin4(x)cos3(x) dx
= ∫(sin4(x)cos2(x) dsin(x)
= ∫(sin4(x)(1-sin2(x)) dsin(x), mental substitution
= ∫(sin4(x)-sin6(x)) dsin(x)
= (1/5)sin5(x) - (1/7)sin7(x) + c


David, cos3(x) dx = cos2(x)*cos(x)dx=cos2(x) d(sin(x))
It is because the derivative of sin(x) is cos(x). Derivative is f′(x)=df/dx, so df=f′(x)dx. Now if f(x)=sin (x) then d(sin(x))=(sin(x))′dx=cos(x) dx. Hop e it helps.
This is mental substitution. You can let u = sin(x)
So, du = cos(x)dx or dsin(x) = cos(x) dx
In general, df(x) is an alternative way of writing f'(x)dx. Thus dsin(x) = (d/dx sin(x)) dx = cos(x) dx.