Question

Evaluate $\int\limits^6_0 {\frac{\, dx }{1+x^{2} } }$ by Trapezoidal rule.

Answer 1

We have to find the integration of

    $\displaystyle \int_{0}^{6}\frac{dx}{1+x^{2}}$

using Trapezoidal rule.

Since there is no mention of number of ordinates, we consider it to be 12.

Solution :

Here: a = x₀ = 0, b = xₙ = 6, n = 11.

∴ h = $\frac{6}{11}$

and y = f(x) = $\frac{1}{1+x^{2}}$

[[ see the given attachment for the computational table ]]

Therefore, $\displaystyle I_{T}^{C}$

= $\frac{h}{2}$ * (Σ Cₙyₙ)

= $\frac{6}{22}$ * 5.2868

= 1.4419

[ Note : The accuracy of this value depends on how many ordinates or intervals, we consider- the more ordinates, the much accurate the value be. ]

•----•

Now, we solve the integration in normal method to find the value solved by Trapezoidal method how much appropriate is.

$\displaystyle \int_{0}^{6}\frac{dx}{1+x^{2}}$

$=\bigg[tan^{-1}x\bigg]_{0}^{6}$

$=tan^{-1}6-tan^{-1}0$

= 1.4056

[ Note : To find the value of arctan in scientific calculator, convert the angle measurements in radian R, not in degree D. ]