Integrate `int_0^(pi/2)f(x)dx` f(x)=cos^3 x*sinx

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use substitution to evaluate the definite integral, hence, you need to come up with the following substitution, such that:

`cos x = t => -sin xdx = dt => sin x dx = (-dt)`

You need to change the limits of intgeration, such that:

`x = 0 => t = cos 0 = 1`

`x = pi/2 => t = cos(pi/2) = 0`

Changing the variable yields:

`int_0^(pi/2) cos^3 x*sin x dx = int_1^0 t^3*(-dt)`

Using th following property of definite integrals yields:

`int_a^b f(x)dx = -int_b^a f(x)dx`

Reasoning by analogy yields:

`int_1^0 t^3*(-dt) = int_0^1 t^3*dt = t^4/4|_0^1`

Using the fundamental theorem of calculus, yields:

`int_0^1 t^3*dt = 1^4/4 - 0^4/4`

`int_0^1 t^3*dt = 1/4`

Hence, evaluating the given definite integral, using the replacement of variable, yields `int_0^(pi/2) cos^3 x*sin x dx = 1/4` .

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