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int igral down 0 up pi  sinx +cosx /(cosx -sinx +201302)dx ?

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sparangello | Student, College Freshman | eNoter

Posted May 20, 2013 at 4:23 PM via web

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int igral down 0 up pi  sinx +cosx /(cosx -sinx +201302)dx ?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted May 20, 2013 at 5:08 PM (Answer #1)

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You need to evaluate the following definite integral, such that:

`int_0^(pi) (sin x + cos x)/(cos x - sin x + 201302) dx`

You should come up with the following substitution, such that:

`cos x - sin x + 201302 = y => (-sin x - cos x) dx = dy`

`- (sin x + cos x) dx = dy => (sin x + cos x) dx = -dy`

You need to change the limits of integration such that:

`x = 0 => cos 0 - sin 0 + 201302 = y => y = 1 + 201302 => y = 201303`

`x = pi => cos pi - sin pi + 201302 = 201302 - 1 = 201301`

Changing the integration variable yields:

`int_201303^201301 (-dy)/y`

You should use the following property of definite integral, such that:

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

Reasoning by analogy yields:

`int_201303^201301 (-dy)/y = int_201301^201303 (dy)/y`

`int_201301^201303 (dy)/y = ln|y||_201301^201303`

Using the fundamental theorem of calculus yields:

`int_201301^201303 (dy)/y = ln 201303 - ln 201301`

`int_201301^201303 (dy)/y = ln (201303/201301)`

Hence, evaluating the given definite integral yields `int_0^(pi) (sin x + cos x)/(cos x - sin x + 201302) dx = ln (201303/201301).`

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