Numerical quadrature by standard rules (trapezoidal, Simpson, Cotes, Chebyshev, Gauss)

Compute by the trapezoidal method the integral $\displaystyle \int_{0}^{1} \dfrac {dx}{1+x}$, taking the number of subintervals $n=4$.

Compute by the trapezoidal method the integral $\displaystyle \int_{1}^{3} \dfrac {dx}{x+1}$, taking the number of subintervals $n=4$.

Compute by the trapezoidal method the integral $\displaystyle \int_{0}^{1} \dfrac {dx}{1+x^{2}}$, taking the number of subintervals $n=4$.

Compute by the trapezoidal method the integral $\displaystyle \int_{0}^{\pi /2} \dfrac {\sin x}{x}\, dx$, taking the number of subintervals $n=4$.

Compute by the trapezoidal method the integral $\displaystyle \int_{0}^{1} \dfrac {\ln (1+x)}{1+x^{2}}\, dx$, taking the number of subintervals $n=6$.

Compute by the trapezoidal method the integral $\displaystyle \int_{0}^{1} \dfrac {\ln (1+x)}{x}\, dx$, taking the number of subintervals $n=6$.

Compute the same integrals as in problems 5223--5228, for $n=4$, by Simpson's formula.

Compute the same integrals as in problems 5223--5228 by Cotes's formula.

Compute the same integrals as in problems 5223--5228 by Chebyshev's method.

Compute the same integrals as in problems 5223--5228 by Gauss's method.

After the substitution $x = \sin \varphi$, compute by Simpson's method with $n=4$ the integrals:

Compute by Simpson's method with $n=4$ the integrals $\displaystyle \int_{0}^{1} \frac{x^{2}\, dx}{\sqrt{1-x^{2}}}$ and $\displaystyle \int_{0}^{1} \frac{dx}{\sqrt{1-x^{4}}}$ with the substitution $1-x=u^{2}$.

After the substitution $x = \dfrac {1}{t}$, compute by Simpson's method with $n=4$ the integral $\displaystyle \int_{1}^{\infty } \frac{x^{2}\, dx}{\sqrt{1+x^{4}}}$.

Using formula 5242 (see also the remark to problem 5243) with $n=4$, compute the integral $\displaystyle \int_{0}^{1} \sqrt{x(1-x^{2})}\, dx$.

Using formula 5240 (see also the remark to problem 5243) with $n=4$, compute the integral $\displaystyle \int_{-1}^{1} \frac{dx}{\sqrt{1-x^{4}}}\, dx$.

Apply Gregory's formula to compute the integral $\displaystyle \int_{0}^{1} \frac{dx}{1+x^{2}}$, taking $n=1$, $h=0{,}1$, carrying out the computation with seven decimal digits.

The same for Gauss's formula.

Carry out an analogous computation (using the Gregory and Gauss formulas, problems 5248--5249) for the integral $\displaystyle \int_{100}^{110} \frac{dx}{x}$.