Find f(x)

This one is a beauty....

The number that you have will be a constant multiple of sin x

so f(x) = k sin x (Where k is a constant!)

Put back in the original equation,

f(x) = \lambda/2 \int_{0}^{\pi/2}{\sin x \sin 2t dt} + sin x \\ \Rightarrow k sin x = k\lambda/2 \sin x \int_{0}^{\pi/2}{\sin 2t dt} + sin x\\ \Rightarrow k = k\lambda/2 \int_{0}^{\pi/2}{\sin 2t dt} + 1\\ \Rightarrow k = k\lambda/2+ 1\\ \Rightarrow k = \frac{2}{2-\lambda}

I mean it is a definite integral... that is why it will be a constant... so you can simply replace it by a constant....

Only sin x and constant will remain...

hence the whole function iwll be of the form k(sin x)

yup that is what i meant :P