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JEE MainMathematicsDefinite Integration

Let f(x) = x^4 ( 10-x 10+x ) . If _ - ^ ( f(x) + x x 1+e^x ) dx = 1 for some (2 , 3 ) , then the value of 2 is equal to _____.

Correct answer

5

Step-by-step solution

Given f(x) = x^4 ( 10-x 10+x ) . Replacing x with -x : f(-x) = (-x)^4 ( 10+x 10-x ) = x^4 ( ( 10-x 10+x )⁻¹ ) = -x^4 ( 10-x 10+x ) = -f(x) Since f(x) is an odd function, its integral over the symmetric interval [- , ] is 0 . _ - ^ f(x) dx = 0 The given equation simplifies to: _ - ^ x x 1+e^x dx = 1 Let h(x) = x x . Since h(-x) = (-x) (-x) = x x , h(x) is an even function. Using the property _ -a ^a h(x) 1+e^x dx = ₀^a h(x) dx for an even function h(x) : ₀^ x x dx = 1 Applying integration by parts: [x (- x)]₀^ - ₀^

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