We are given the definite integral
[math]I = \displaystyle \int_0^{\pi/4} \cos^{\frac{3}{2}}(2x) \cos{x} \, dx. \tag*{}[/math]
Using the double angle identity for cosine and then make the substitution [math]t = \sqrt{2} \sin{x}[/math], we obtain
[math]\begin{align*} I &= \displaystyle \int_0^{\pi/4} (1 - 2 \sin^2{x})^{\frac{3}{2}} \cos{x} \, dx\\ &= \int_0^1 (1 - t^2)^{\frac{3}{2}} \cdot \frac{1}{\sqrt{2}} \, dt\\ &= \frac{1}{\sqrt{2}} \int_0^1 (1 - t^2)^{\frac{3}{2}} \, dt. \end{align*} \tag*{}[/math]
Next, we let [math]w = t^2[/math] to convert the integral into a s...
Không có nhận xét nào:
Đăng nhận xét