int_0^{frac{pi}{2}} frac{sin^2 x}{sin x + cos | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) $I = \int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx \quad ... (i)$ગુણધર્મ $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ નો ઉપયોગ કરતા:$I = \int_0^{\

Vedclass · Accepted Answer

$\frac{1}{\sqrt{2}} \log(\sqrt{2} + 1)$

Vedclass · Answer

(B) $I = \int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx \quad ... (i)$ગુણધર્મ $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ નો ઉપયોગ કરતા:$I = \int_0^{\pi/2} \frac{\cos^2 x}{\cos x + \sin x} dx \quad ... (ii)$$(i)$ અને $(ii)$ નો સરવાળો કરતા:$2I = \int_0^{\pi/2} \frac{\sin^2 x + \cos^2 x}{\sin x + \cos x} dx = \int_0^{\pi/2} \frac{1}{\sin x + \cos x} dx$$2I = \frac{1}{\sqrt{2}} \int_0^{\pi/2} \frac{1}{\sin(x + \pi/4)} dx = \frac{1}{\sqrt{2}} \int_0^{\pi/2} \operatorname{cosec}(x + \pi/4) dx$$2I = \frac{1}{\sqrt{2}} [\log|\operatorname{cosec}(x + \pi/4) - \cot(x + \pi/4)|]_0^{\pi/2}$$2I = \frac{1}{\sqrt{2}} [\log|\operatorname{cosec}(3\pi/4) - \cot(3\pi/4)| - \log|\operatorname{cosec}(\pi/4) - \cot(\pi/4)|]$$2I = \frac{1}{\sqrt{2}} [\log(\sqrt{2} + 1) - \log(\sqrt{2} - 1)] = \frac{1}{\sqrt{2}} \log\left(\frac{\sqrt{2} + 1}{\sqrt{2} - 1}\right)$કારણ કે $\frac{\sqrt{2} + 1}{\sqrt{2} - 1} = (\sqrt{2} + 1)^2$, તેથી $2I = \frac{1}{\sqrt{2}} \log(\sqrt{2} + 1)^2 = \frac{2}{\sqrt{2}} \log(\sqrt{2} + 1)$$I = \frac{1}{\sqrt{2}} \log(\sqrt{2} + 1)$

$\int_0^{\frac{\pi}{2}} \frac{\sin^2 x}{\sin x + \cos x} dx = $

Similar Questions

નિશ્ચિત સંકલનનું મૂલ્ય શોધો: $\int_{0}^{\infty} \frac{x}{(1 + x)(1 + x^2)} \, dx$

જો $\int_{-1}^{4} f(x) dx = 4$ અને $\int_{2}^{4} (3 - f(x)) dx = 7$ હોય,તો $\int_{2}^{-1} f(x) dx$ ની કિંમત શોધો.

$\sum_{n=1}^{100} \int_{n-1}^{n} e^{x-[x]} dx$ નું મૂલ્ય શોધો,જ્યાં $[x]$ એ $x$ થી નાનો અથવા તેના જેટલો મહત્તમ પૂર્ણાંક છે.

$\int_{0}^{1} \frac{8 \log(1+x)}{1+x^{2}} dx = $

$\int_0^{\frac{\pi}{2}} \left( \frac{\sqrt[n]{\sec x}}{\sqrt[n]{\sec x} + \sqrt[n]{\operatorname{cosec} x}} \right) dx = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module