Newton Raphson For $\int_{0}^{x} \frac{\sin t}{1+t}dt=0.25$

Question

1. Write the general iteration of Newton Raphson for $$\int_{0}^{x} \frac{\sin t}{1+t}dt=0.25$$

2. Run one iteration with $x_0=1.5$, Approximate the integral using the trapezoidal rule with $4$ intervals

1. We basically need to write the following:

$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$

So the answer is:

$$x_{n+1}=x_n -\frac{\int_{0}^{x_n}\frac{\sin t}{1+t}dt-0.25}{\frac{\sin x_n}{1+x_n}}?$$

2. we will first approximate $\int_{0}^{1.5} \frac{\sin t}{1+t}dt$

$$\int_{0}^{1.5} \frac{\sin t}{1+t}dt\approx \frac{\frac{1.5}{4}}{2}( \frac{\sin 0}{1+}+2[ \frac{\sin 0.375}{1+0.37}+ \frac{\sin 0.7}{1+0.7}+ \frac{\sin 1.125}{1+}]+\frac{\sin 1.5}{1+1.5})=\\=\frac{0.375}{2}(0+2[0.266+0.389+0.424]+0.398)=0.479$$

So $$x_{1}=1.5 -\frac{0.479-0.25}{\frac{\sin 1.5}{1+1.5}}=0.926?$$