Mathematics

If $$\int _{ 0 }^{ a }{ f\left( x \right) dx=\int _{ 0 }^{ a }{ f\left( a-x \right) dx }  } $$, then the value of $$\int _{ 0 }^{ \pi  }{ xf\left( \sin { x }  \right)  } dx=$$


ANSWER

$$\dfrac { \pi }{ 2 } \int _{ 0 }^{ \pi }{ f\left( \sin { x } \right) dx } $$


SOLUTION
Let $$S=\int_{0}^{π} xf(sinx)dx$$
$$\int_{0}^{π} xf(sinx) dx=\int_{0}^{π} (π-x)f(sin(π-x))dx$$
                        $$S=\int_{0}^{π}(π-x)f(sinx)dx$$
                        $$S=\int_{0}^{π}πf(sinx)dx-\int_{0}^{π}xf(sinx)dx$$
                        $$S=π\int_{0}^{π}f(sinx)dx-S$$
                       $$2S=π\int_{0}^{π}f(sinx)dx$$
                        $$S=\dfrac{π}{2}\int_{0}^{π}f(sinx)dx$$
$$\int_{0}^{π} xf(sinx) dx=\dfrac{π}{2}\int_{0}^{π}f(sinx)dx$$
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Single Correct Medium Published on 17th 09, 2020
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