IGNOU MPH 14 2026 - Evaluate the integral using the 2-point Gaussian Quadrature

Question

Evaluate the integral $\int_{1}^{2} x^3 \ln x \, dx$ using the 2-point Gaussian Quadrature Rule with the following values:

Points	Weights `(C_i)`	Arguments `(x_i`)
1	1.0	0.577
2	1.0	`−0.577`

19 Jan 2026

Answer :

Word Count : 358

We are asked to evaluate [ \int_{1}^{2} x^3 \ln x , dx ] using a 2-point Gaussian Quadrature. Let's solve this step by step manually. --- Step 1: Transform the interval The standard 2-point Gaussian Quadrature formula is for the interval ([-1, 1]): [ \int_{-1}^{1} f(t), dt \approx \sum_{i=1}^{2} C_i f(t_i) ] where (C_1 = C_2 = 1), (t_1 = 0.577), (t_2 = -0.577). For the interval ([a, b] = [1, 2]), we use the linear transformation: [ x = \frac{b-a}{2} t + \frac{a+b}{2} = \frac{2-1}{2} t + \frac{1+2}{2} = \frac{1}{2} t + \frac{3}{2} ] and [ dx = \frac{b-a}{2} dt = \frac{1}{2} dt ] So the integral becomes: [ \int_{1}^{2} x^3 \ln x , dx = \frac{1}{2} \int_{-1}^{1} \left( \frac{1}{2} t + \frac{3}{2} \right)^3 \ln \left( \frac{1}{2} t + \frac{3}{2} \right) dt ] --- Step 2: Apply 2-point Gaussian Quadrature The formula

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