Find Antiderivative of x^2: We need to find the definite integral of x^2 from 1 to 2. The first step is to find the antiderivative of x^2. The antiderivative of x^n, where n is a real number and n ≠ -1, is (x^(n+1))/(n+1) + C, where C is the constant of integration. For x^2, the antiderivative is (x^(2+1))/(2+1) + C, which simplifies to (x^3)/3 + C.Evaluate Definite Integral: Now that we have the antiderivative, we can evaluate the definite integral from 1 to 2. We will apply the Fundamental Theorem of Calculus, which states that if F is an antiderivative of f on an interval [a, b], then the definite integral of f from a to b is F(b) - F(a). So, we need to calculate ((2^3)/3) - ((1^3)/3).Perform Calculations: Let's perform the calculations: For x = 2: (2^3)/3 = (8)/3. For x = 1: (1^3)/3 = (1)/3. Now, subtract the value at x = 1 from the value at x = 2: (8/3) - (1/3) = (7/3).Final Answer: The final answer is the result of the subtraction, which is 7/3. This is the value of the definite integral of x^2 from 1 to 2.

Resources

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$3 \int_{1}^{2} x^{2} d x$

View step-by-step help

Home

Math Problems

Algebra 1

Evaluate integers raised to rational exponents

Full solution

3 \int_{1}^{2} x^{2} d x

Find Antiderivative of $x^2$ : We need to find the definite integral of $x^2$ from $1$ to $2$ . The first step is to find the antiderivative of $x^2$ . The antiderivative of $x^n$ , where $n$ is a real number and $n \neq -1$ , is $\frac{x^{(n+1)}}{(n+1)} + C$ , where $C$ is the constant of integration. For $x^2$ , the antiderivative is $x^2$ $1$ , which simplifies to $x^2$ $2$ .
Evaluate Definite Integral: Now that we have the antiderivative, we can evaluate the definite integral from $1$ to $2$ . We will apply the Fundamental Theorem of Calculus, which states that if $F$ is an antiderivative of $f$ on an interval $[a, b]$ , then the definite integral of $f$ from $a$ to $b$ is $F(b) - F(a)$ . So, we need to calculate $\left(\frac{2^3}{3}\right) - \left(\frac{1^3}{3}\right)$ .
Perform Calculations: Let's perform the calculations: $\newline$ For $x = 2$ : $\frac{2^3}{3} = \frac{8}{3}$ . $\newline$ For $x = 1$ : $\frac{1^3}{3} = \frac{1}{3}$ . $\newline$ Now, subtract the value at $x = 1$ from the value at $x = 2$ : $\frac{8}{3} - \frac{1}{3} = \frac{7}{3}$ .
Final Answer: The final answer is the result of the subtraction, which is $\frac{7}{3}$ . This is the value of the definite integral of $x^2$ from $1$ to $2$ .