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JEE MainMathematicsDefinite Integration

The value of 10 ₀^ /2 ( x - 2x^2) (2x) 3^ x 3^ x + 3^ x dx is equal to

Correct answer

5

Step-by-step solution

Let I = ₀^ /2 ( x - 2x^2) (2x) 3^ x 3^ x + 3^ x dx Using the property ₀^ a f(x) dx = ₀^ a f(a-x) dx , we replace x with 2 - x : I = ₀^ /2 ( ( 2 - x ) - 2 ( 2 - x )^2 ) (2 ( 2 - x ) ) 3^ ( /2 - x) 3^ ( /2 - x) + 3^ ( /2 - x) dx Simplifying the terms: Polynomial part: ^2 2 - x - 2 ( ^2 4 - x + x^2 ) = x - 2x^2 Trigonometric part: ( - 2x) = (2x) Exponential part: 3^ x 3^ x + 3^ x So, I = ₀^ /2 ( x - 2x^2) (2x) 3^ x 3^ x + 3^ x dx Adding the two equations for I : 2I = ₀^ /2 ( x - 2x^2) (2x) ( 3^ x + 3^ x 3^ x + 3^ x )

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