Given the integrals A and B, find A+B? A=Integral x*cos^2 xdx B = Integral x*sin^2x dx

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have the integrals A and B defined as A = Int[x*(cos x)^2 dx]

and B = Int[x*(sin x)^2 dx]

A + B

=> Int[x*(cos x)^2 dx] + Int[x*(sin x)^2 dx]

=> Int[x*(cos x)^2 + x*(sin x)^2 dx]

=> Int[x*((cos x)^2 + (sin x)^2) dx]

use the property that (sin x)^2 + (cos x)^2 = 1

=> Int[x dx]

=> x^2 / 2 + C

The value of A + B = x^2/2 + C

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll calculate the sum of the given integrals, using the property of integral to be additive.

Int x*(cos x)^2 dx + Int x*(sin x)^2 dx = Int x*[(cos x)^2 + (sin x)^2] dx

We'll use the Pythagorean identity:

(cos x)^2 + (sin x)^2 = 1

A + B = Int x*[(cos x)^2 + (sin x)^2] dx = Int x dx

A + B = Int x dx

A + B = x^2/2 + C

The requested sum of the given integrals is A+B = x^2/2 + C.

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