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Question
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
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Solution
Consider the given integral
`I= int_(-2)^2x^2/(1+5^x)dx`
Let us use the property,
`int_a^bf(x)dx=int_b^af(a+b-x)dx`
`:.I = int_(-2)^2(-x)^2/(1+5^(-x))dx`
`=int_(-2)^2(5^(x)x^2)/(1+5^x)dx `
Adding equations (1) and (2), we have,
`2I=int_(-2)^2(1+5^x)/(1+5^x)xx x^2dx`
`=int_(-2)^2x^2dx`
`=[x^3/3]^2`
`=1/3[8-(8)]`
`=1/3[16]`
`=>I= 8/3`
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