Question Statement

Evaluate the definite integral:

$\int_{- 1}^{2} (2 x + 3) d x$

Background and Explanation

This problem involves computing a definite integral using the linearity property of integration, which allows us to integrate term by term. You will need the power rule for integration and the Fundamental Theorem of Calculus to evaluate the antiderivative at the given bounds.

Solution

We begin by applying the linearity property of definite integrals, which states that $\int (a f (x) + b g (x)) d x = a \int f (x) d x + b \int g (x) d x$ . This allows us to separate the integral into two simpler parts:

$\int_{- 1}^{2} (2 x + 3) d x = 2 \int_{- 1}^{2} x d x + 3 \int_{- 1}^{2} 1 d x$

Next, we find the antiderivative of each term using the power rule $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1}$ :

The antiderivative of $x$ is $\frac{x ^{2}}{2}$
The antiderivative of $1$ is $x$

Applying the Fundamental Theorem of Calculus, we evaluate these antiderivatives at the upper bound ( $2$ ) and lower bound ( $- 1$ ):

$= 2 [\frac{x ^{2}}{2}]_{- 1}^{2} + 3 [x]_{- 1}^{2}$

In the first term, the coefficient $2$ and the denominator $2$ cancel each other:

$= [x^{2}]_{- 1}^{2} + 3 [x]_{- 1}^{2}$

Now we substitute the bounds into each expression. For the first term: $(2)^{2} - (- 1)^{2}$ . For the second term: $3 \times (2 - (- 1))$ :

$= ((2)^{2} - (- 1)^{2}) + 3 (2 - (- 1))$

Simplifying the arithmetic step by step:

$= (4 - 1) + 3 (2 + 1)$ $= 3 + 3 (3)$ $= 3 + 9$ $= 12$

Therefore, the value of the definite integral is 12.

Key Formulas or Methods Used

Linearity of Integration: $\int_{a}^{b} [c f (x) + d g (x)] d x = c \int_{a}^{b} f (x) d x + d \int_{a}^{b} g (x) d x$
Power Rule: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1}$ (for $n \neq = - 1$ )
Fundamental Theorem of Calculus: $\int_{a}^{b} f (x) d x = F (b) - F (a)$ , where $F (x)$ is the antiderivative of $f (x)$

Summary of Steps

Separate the integral using linearity: $2 \int_{- 1}^{2} x d x + 3 \int_{- 1}^{2} 1 d x$
Integrate each term: Find antiderivatives $\frac{x ^{2}}{2}$ and $x$
Apply FTC (Fundamental Theorem of Calculus) and evaluate at bounds $2$ and $- 1$
Simplify: $(4 - 1) + 3 (3) = 3 + 9 = 12$

Solution

We begin by applying the linearity property of definite integrals, which states that

\int (a f (x) + b g (x)) d x = a \int f (x) d x + b \int g (x) d x

. This allows us to separate the integral into two simpler parts:

\int_{- 1}^{2} (2 x + 3) d x = 2 \int_{- 1}^{2} x d x + 3 \int_{- 1}^{2} 1 d x

Next, we find the antiderivative of each term using the power rule

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1}

The antiderivative of

x

\frac{x ^{2}}{2}

The antiderivative of

1

x

Applying the Fundamental Theorem of Calculus, we evaluate these antiderivatives at the upper bound (

2

) and lower bound (

- 1

= 2 [\frac{x ^{2}}{2}]_{- 1}^{2} + 3 [x]_{- 1}^{2}

In the first term, the coefficient

2

and the denominator

2

cancel each other:

= [x^{2}]_{- 1}^{2} + 3 [x]_{- 1}^{2}

Now we substitute the bounds into each expression. For the first term:

(2)^{2} - (- 1)^{2}

. For the second term:

3 \times (2 - (- 1))

= ((2)^{2} - (- 1)^{2}) + 3 (2 - (- 1))

Simplifying the arithmetic step by step:

= (4 - 1) + 3 (2 + 1)

= 3 + 3 (3)

= 3 + 9

= 12

Therefore, the value of the definite integral is 12.

3.7 Q-1

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

3.7 Q-1

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps