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Let $I_{1} = \int_{a}^{b} (\int_{a}^{x} f (t) d t) d x$ and $I_{2} = \int_{a}^{b} (b - x) f (x) d x$ then

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I1=I2

I1=(b−a)+I2

I1=(b−a)I2

none of these

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detailed solution

Correct option is A

Let F(x)=∫αx f(t)dt then F′(x)=f(x)I1=∫ab F(x)dx=xF(x)ab−∫ab xf(x)dx=bF(b)-aFa−∫ab xf(x)dx=∫ab (b−x)f(x)dx=I2.

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Let I1=∫ab ∫ax f(t)dtdx and I2=∫ab (b−x)f(x) dx then

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Let $I_{1} = \int_{a}^{b} (\int_{a}^{x} f (t) d t) d x$ and $I_{2} = \int_{a}^{b} (b - x) f (x) d x$ then