3.6 Q-1

Question Statement

Evaluate the integral:
$\int_{1}^{2}\left(x^{2}+1\right) , dx$

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Background and Explanation

To solve this problem, we need to apply the basic rules of integration. The integral consists of two parts:

• The integral of $x^2$
• The integral of $1$

We will use the power rule for the first term and the standard result for the second term. The power rule is: $\int x^n , dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for any integer n)}$

The integral of 1 with respect to $x$ is simply $x$. These concepts will help us break down and solve the integral step by step.

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Solution

1. Separate the Integral:

   We begin by separating the given integral into two simpler integrals: $\int_{1}^{2} \left( x^2 + 1 \right) , dx = \int_{1}^{2} x^2 , dx + \int_{1}^{2} 1 , dx$

2. Integrate $x^2$:

   Using the power rule: $\int x^2 , dx = \frac{x^3}{3}$

   Now, apply the limits of integration from 1 to 2: $\int_{1}^{2} x^2 , dx = \left[ \frac{x^3}{3} \right]_{1}^{2} = \frac{2^3}{3} - \frac{1^3}{3} = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}$

3. Integrate 1:

   The integral of 1 is straightforward: $\int 1 , dx = x$

   Applying the limits of integration from 1 to 2: $\int_{1}^{2} 1 , dx = \left[ x \right]_{1}^{2} = 2 - 1 = 1$

4. Combine the Results:

   Adding the two results together: $\frac{7}{3} + 1 = \frac{7}{3} + \frac{3}{3} = \frac{10}{3}$

Thus, the value of the integral is: $\boxed{\frac{10}{3}}$

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Key Formulas or Methods Used

• Power rule of integration:
  $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$
• Basic integral of 1:
  $\int 1 , dx = x$

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Summary of Steps

1. Split the integral into two parts:
   $\int_{1}^{2} (x^2 + 1) , dx = \int_{1}^{2} x^2 , dx + \int_{1}^{2} 1 , dx$

2. Integrate $x^2$ and apply limits:
   $\int_{1}^{2} x^2 , dx = \frac{7}{3}$

3. Integrate 1 and apply limits:
   $\int_{1}^{2} 1 , dx = 1$

4. Add the results:
   $\frac{7}{3} + 1 = \frac{10}{3}$

Thus, the final result is $\frac{10}{3}$.