Evaluate : 0^1 1+x^3\ dx , using Simpson's 13 rule with n=4 . [6] OR Evaluate : 0^1 dx1+x , using Trapezoidal rule for n=4 .

Solution 1: Using Simpson’s 1/3 Rule for ${\int }_0^1\sqrt{1+x^3}dx$ with $n=4$

Step 1: Understand the formula

Simpson’s 1/3 rule for numerical integration is given by:

$${\int }_a^{b}f(x)dx\approx \frac{h}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+f(x_4)\right]$$

where:

• $h=\frac{b-a}{n}$ (step size),
• $n$ must be even (here $n=4$),
• $x_i=a+ih$ for $i=0,1,2,3,4$.

Step 2: Compute $h$ and $x_i$ values

Given:

• $a=0$, $b=1$, $n=4$.

$$h=\frac{1-0}{4}=0.25$$

Compute $x_i$: $$\begin{array}{rl}{x}_0& =0+0\times 0.25=0\\ x_1& =0+1\times 0.25=0.25\\ x_2& =0+2\times 0.25=0.5\\ x_3& =0+3\times 0.25=0.75\\ x_4& =0+4\times 0.25=1\end{array}$$

Step 3: Evaluate $f(x_i)=\sqrt{1+x_i^3}$

$$\begin{array}{rl}f(x_0)& =\sqrt{1+0^3}=\sqrt{1}=1\\ f(x_1)& =\sqrt{1+(0.25{)}^3}=\sqrt{1+0.015625}=\sqrt{1.015625}\approx 1.0078\\ f(x_2)& =\sqrt{1+(0.5{)}^3}=\sqrt{1+0.125}=\sqrt{1.125}\approx 1.0607\\ f(x_3)& =\sqrt{1+(0.75{)}^3}=\sqrt{1+0.421875}=\sqrt{1.421875}\approx 1.1923\\ f(x_4)& =\sqrt{1+1^3}=\sqrt{2}\approx 1.4142\end{array}$$

Step 4: Apply Simpson’s 1/3 rule

$${\int }_0^1\sqrt{1+x^3}dx\approx \frac{0.25}{3}\left[1+4(1.0078)+2(1.0607)+4(1.1923)+1.4142\right]$$

Calculate the sum inside the brackets: $$1+4(1.0078)+2(1.0607)+4(1.1923)+1.4142=1+4.0312+2.1214+4.7692+1.4142=13.336$$

Now multiply by $\frac{h}{3}=\frac{0.25}{3}$: $$\frac{0.25}{3}\times 13.336\approx 1.1097$$

Final Answer

$$\boxed{1.1097}$$

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Solution 2: Using Trapezoidal Rule for ${\int }_0^1\frac{dx}{1+x}$ with $n=4$

Step 1: Understand the formula

The trapezoidal rule is given by:

$${\int }_a^{b}f(x)dx\approx \frac{h}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+f(x_4)\right]$$

where:

• $h=\frac{b-a}{n}$,
• $x_i=a+ih$.

Step 2: Compute $h$ and $x_i$ values

Given:

• $a=0$, $b=1$, $n=4$.

$$h=\frac{1-0}{4}=0.25$$

Compute $x_i$: $$\begin{array}{rl}{x}_0& =0\\ x_1& =0.25\\ x_2& =0.5\\ x_3& =0.75\\ x_4& =1\end{array}$$

Step 3: Evaluate $f(x_i)=\frac{1}{1+x_i}$

$$\begin{array}{rl}f(x_0)& =\frac{1}{1+0}=1\\ f(x_1)& =\frac{1}{1+0.25}=\frac{1}{1.25}=0.8\\ f(x_2)& =\frac{1}{1+0.5}=\frac{1}{1.5}\approx 0.6667\\ f(x_3)& =\frac{1}{1+0.75}=\frac{1}{1.75}\approx 0.5714\\ f(x_4)& =\frac{1}{1+1}=0.5\end{array}$$

Step 4: Apply the trapezoidal rule

$${\int }_0^1\frac{dx}{1+x}\approx \frac{0.25}{2}\left[1+2(0.8)+2(0.6667)+2(0.5714)+0.5\right]$$

Calculate the sum inside the brackets: $$1+2(0.8)+2(0.6667)+2(0.5714)+0.5=1+1.6+1.3334+1.1428+0.5=5.5762$$

Now multiply by $\frac{h}{2}=\frac{0.25}{2}=0.125$: $$0.125\times 5.5762\approx 0.6970$$

Final Answer

$$\boxed{0.6970}$$