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प्रश्न
Evaluate the following integrals using properties of integration:
`int_(- pi/2)^(pi/2) (x^5 + x cos x + tan^3 x + 1) "d"x`
उत्तर
`int_(- pi/2)^(pi/2) (x^5 + x cos x + tan^3 x) "d"x`
= `int_((-pi)/2)^(pi/2) (x^5 + x cos x + tan^3x) "d"x + int_((- pi)/2)^(pi/2)`
= Let f(x) = x5 + x cos x + tan3x
f(– x) = – x5 – x cos x – tan3x
f(x) = – f(– x)
f(x) is an odd function
∴ `int_((- pi)/2)^(pi/2) (x^5 + x cos x + tan^3x) "d"x` = 0
Let g(x) = `int_((-pi)/2)^(pi/2) "d"x = [x]_((- pi)/2)^(pi/2)`
= `pi/2 - (- pi/2)`
= `pi`
`int_(- pi/2)^(pi/2) (x^5 + x cos x + tan^3 x) "d"x`
= `int_((-pi)/2)^(pi/2) f(x)"d"x + int_((-pi)/2)^(pi/2) "g"(x) "d"x`
= `0 + pi`
= `pi`
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