Evaluate the definite integral $\int_0^{\pi/2} \sin^3(x) \cos(x) dx$.
Let u = sin x, du = cos x dx. Integral becomes $\int_0^1 u^3 du = [u^4/4]_0^1 = 1/4 = 0.25$.
Evaluate $\int_0^1 \int_0^x (x + y) dy dx$.
Inner integral: [xy + $y^2/2$]$_0^x$ = $x^2 + x^2/2$ = 1.5 $x^2$. Outer integral: $\int_0^1 1.5 x^2 dx = [0.5 x^3]_0^1 = 0.5$.
The improper integral $\int_1^{\infty} \frac{1}{x^p} dx converges$ if and only if:
Standard p-test for improper integrals: $\int_1^{\infty} x^{-p} dx converges$ to 1/(p-1) for p > 1 and diverges for p <= 1.
Calculate the value of Gamma function $\Gamma(5/2)$. (Use $\pi \approx 3.1416)$
$\Gamma(5/2) = (3/2)(1/2)\Gamma(1/2) = \frac{3}{4} \sqrt{\pi} \approx 0.75 * 1.77245 = 1.3293 \approx 1.33$.
Evaluate $\int_{-\infty}^{\infty} e^{-x^2} dx$.
Standard Gaussian integral value is $\sqrt{\pi} \approx 1.77245$. Rounding to 1.77.
