Mathematics

Evaluate : 
$$\int\limits_0^{\pi /2} {\dfrac{{\cos x\,dx}}{{\left( {\,\cos \,x\, + \,\sin x} \right)}}}  $$


SOLUTION
We have,
$$I=\int\limits_0^{\pi /2} {\dfrac{{\cos x\,dx}}{{\left( {\,\cos \,x\, + \,\sin x} \right)}}}  $$              $$.........(1)$$

We know that
$$\int\limits_b^{a}f(x)dx=\int \limits_b^a\ f(a+b-x)dx$$

Therefore,
$$I=\int\limits_0^{\pi /2} {\dfrac{{\sin x\,dx}}{{\left( {\,\sin \,x\, + \,\cos x} \right)}}}  $$             $$............(2)$$

On adding equations $$(1)$$ and $$(2)$$, we get
$$2I=\int\limits_0^{\pi /2} {\dfrac{{\sin x+\cos x}}{{\left( {\,\sin \,x\, + \,\cos x} \right)}}} dx$$

$$2I=\int\limits_0^{\pi /2} 1 dx$$

$$2I=[x]_0^\frac{\pi}{2}$$

$$2I=\dfrac{\pi}{2}-0$$

$$2I=\dfrac{\pi}{2}$$

$$I=\dfrac{\pi}{4}$$

Hence, this is the answer.
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