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Prove that:int(-a)^(a) f(x) dx = {{:(2in...

Prove that:`int_(-a)^(a) f(x) dx = {{:(2int_(0)^(a)f(x)dx, f(x) " is even "),(0, f(x) " is odd"):}` and hence Evaluate `int_(-t)^(t) sin^(5)(x)cos^(4)(x) dx`

Text Solution

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The correct Answer is:
`int_(-1)^(1) sin^(5)xcos^(4)x dx=0`
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Prove that int_(-a)^(a) dx = {(2int_(0)^(a) f(x) dx, if f(x) "is even"),(0, if f(x) "is odd"):} and hence evaluate (a) int_(-1)^(1) sin^(5)x cos^(4)xdx .

Prove that int_(-a)^(a) dx = {(2int_(0)^(a) f(x) dx, if f(x) "is even"),(0, if f(x) "is odd"):} and hence evaluate (c) int_(0)^(pi)|cosx|dx .

Knowledge Check

  • If int_(-1)^(4) f(x) dx = 4 and int_(2)^(4) f(x) dx = 7 then int_2^(-1) f(x) dx =

    A
    3
    B
    minus 5
    C
    2
    D
    4

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    Prove that int_(-a)^(a) dx = {(2int_(0)^(a) f(x) dx, if f(x) "is even"),(0, if f(x) "is odd"):} and hence evaluate (b) int_(-pi//2)^(pi//2) sin^(7) x dx .

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