Question Statement

Evaluate the definite integral:

$\int_{0}^{3} (6 x^{2} - 4 x + 5) d x$

Background and Explanation

This problem requires computing a definite integral of a polynomial function. You'll apply the linearity property to break the integral into simpler parts, use the power rule to find antiderivatives, and then apply the Fundamental Theorem of Calculus to evaluate at the bounds $x = 0$ and $x = 3$ .

Solution

We solve this step-by-step using the properties of definite integrals.

Step 1: Use linearity to split the integral into three separate terms.

$\int_{0}^{3} (6 x^{2} - 4 x + 5) d x = 6 \int_{0}^{3} x^{2} d x - 4 \int_{0}^{3} x d x + 5 \int_{0}^{3} 1 d x$

Step 2: Apply the power rule $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1}$ to find each antiderivative, multiplying by the respective coefficients.

For the first term: $6 \cdot \frac{x ^{3}}{3} = 2 x^{3}$
For the second term: $- 4 \cdot \frac{x ^{2}}{2} = - 2 x^{2}$
For the third term: $5 \cdot x = 5 x$

This gives us:

$= [2 x^{3}]_{0}^{3} - [2 x^{2}]_{0}^{3} + [5 x]_{0}^{3}$

Step 3: Evaluate each antiderivative at the upper limit ( $x = 3$ ) and lower limit ( $x = 0$ ), then subtract.

$= 2 ((3)^{3} - (0)^{3}) - 2 ((3)^{2} - (0)^{2}) + 5 (3 - 0)$

Step 4: Simplify the arithmetic inside each parentheses.

$= 2 (27 - 0) - 2 (9 - 0) + 5 (3)$

$= 2 (27) - 2 (9) + 15$

Step 5: Perform the final multiplication and addition/subtraction.

$= 54 - 18 + 15$

$= 51$

Therefore, the value of the definite integral is 51.

Key Formulas or Methods Used

Linearity of Integration: $\int [a f (x) + b g (x)] d x = a \int f (x) d x + b \int g (x) d x$
Power Rule for Integration: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ (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

Split the integral into three separate integrals using linearity: $6 \int_{0}^{3} x^{2} d x - 4 \int_{0}^{3} x d x + 5 \int_{0}^{3} 1 d x$
Integrate each term using the power rule, simplifying coefficients: $2 x^{3}$ , $- 2 x^{2}$ , and $5 x$
Evaluate each antiderivative at the bounds $x = 3$ and $x = 0$
Calculate the values: $2 (27) - 2 (9) + 5 (3) = 54 - 18 + 15$
Simplify to obtain the final result: $51$

Question Statement

Evaluate the definite integral:

$\int_{0}^{3} (6 x^{2} - 4 x + 5) d x$

Background and Explanation

Solution

We solve this step-by-step using the properties of definite integrals.

Step 1: Use linearity to split the integral into three separate terms.

$\int_{0}^{3} (6 x^{2} - 4 x + 5) d x = 6 \int_{0}^{3} x^{2} d x - 4 \int_{0}^{3} x d x + 5 \int_{0}^{3} 1 d x$

Step 2: Apply the power rule $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1}$ to find each antiderivative, multiplying by the respective coefficients.

For the first term: $6 \cdot \frac{x ^{3}}{3} = 2 x^{3}$
For the second term: $- 4 \cdot \frac{x ^{2}}{2} = - 2 x^{2}$
For the third term: $5 \cdot x = 5 x$

This gives us:

$= [2 x^{3}]_{0}^{3} - [2 x^{2}]_{0}^{3} + [5 x]_{0}^{3}$

Step 3: Evaluate each antiderivative at the upper limit ( $x = 3$ ) and lower limit ( $x = 0$ ), then subtract.

$= 2 ((3)^{3} - (0)^{3}) - 2 ((3)^{2} - (0)^{2}) + 5 (3 - 0)$

Step 4: Simplify the arithmetic inside each parentheses.

$= 2 (27 - 0) - 2 (9 - 0) + 5 (3)$

$= 2 (27) - 2 (9) + 15$

Step 5: Perform the final multiplication and addition/subtraction.

$= 54 - 18 + 15$

$= 51$

Therefore, the value of the definite integral is 51.

Key Formulas or Methods Used

Linearity of Integration: $\int [a f (x) + b g (x)] d x = a \int f (x) d x + b \int g (x) d x$
Power Rule for Integration: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ (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

Split the integral into three separate integrals using linearity: $6 \int_{0}^{3} x^{2} d x - 4 \int_{0}^{3} x d x + 5 \int_{0}^{3} 1 d x$
Integrate each term using the power rule, simplifying coefficients: $2 x^{3}$ , $- 2 x^{2}$ , and $5 x$
Evaluate each antiderivative at the bounds $x = 3$ and $x = 0$
Calculate the values: $2 (27) - 2 (9) + 5 (3) = 54 - 18 + 15$
Simplify to obtain the final result: $51$

3.7 Q-4

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

3.7 Q-4

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps