Question

Evaluate the following integrals using properties of integration:

`int_(- pi/2)^(pi/2) (x^5 + x cos x + tan^3 x + 1)  "d"x`

Answer 1

`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`