Question

If $\Phi \left(x\right)=\int \frac{dx}{si{n}^{\frac{1}{2}}xco{s}^{\frac{7}{2}}x}$, then $\Phi \left(\frac{\pi }{4}\right)-\Phi \left(0\right)=$

• A. $\frac{12}{5}$
• B. $\frac{9}{5}$
• C. $\frac{6}{5}$
• D. \N

Answer 1

The correct option is A $\frac{12}{5}$

$tanx=t\Rightarrow se{c}^{2}xdx=dt\therefore f\left(x\right)=\int \frac{dx}{\frac{si{n}^{\frac{1}{2}}x}{co{s}^{\frac{1}{2}}x}.co{s}^{4}x}=\int \frac{(1+ta{n}^{2}x)se{c}^{2}x}{\sqrt{tanx}}dx=\int \frac{(1+t^2)}{\sqrt{t}}dt=\int (t^{-1/2}+t^{3/2})dt=2t^{1/2}+\frac{2}{5}{t}^{5/2}=2\sqrt{tanx}+\frac{2}{5}(tanx{)}^{5/2}\therefore \varphi \left(\frac{\pi }{4}\right)-\varphi (0)=2+\frac{2}{5}=\frac{12}{5}$