Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{2}\left(3 x^{2}+2 x\right) d x$$

Short answer

Answer: The value of the definite integral is 12.

Step-by-step solution

Find the antiderivative of each term

Use the power rule for integration to find the antiderivative of each term: $$\int 3x^2 dx = x^3 + C_1$$ $$\int 2x dx = x^2 + C_2$$ Combine the antiderivatives to form the complete antiderivative function: $$\int (3x^2 + 2x) dx = x^3 + x^2 + C$$

Apply the Fundamental Theorem of Calculus

Now that we have the antiderivative function, we can apply the Fundamental Theorem of Calculus by evaluating the function at the upper and lower limits of integration and taking the difference: $$\int_{0}^{2} \left(3x^2 + 2x \right)dx = (x^3 + x^2) \Big|_0^2$$

Evaluate the antiderivative function at the limits of integration

Substitute the upper limit (2) and the lower limit (0) into the antiderivative function, then take the difference: $$(2^3 + 2^2) - (0^3 + 0^2) = (8 + 4) - (0 + 0) = 12$$

Write the final answer

The definite integral of the function $$3x^2 + 2x$$ from 0 to 2 is: $$\int_{0}^{2} \left(3x^2 + 2x \right)dx = 12$$

Key concepts

Definite Integral

A definite integral is a powerful mathematical tool used to calculate the accumulated quantity between two points along a continuous curve. When you encounter a definite integral like \( \int_{0}^{2} (3x^2 + 2x) \, dx \), it represents the area under the curve of the function \(3x^2 + 2x\) from \(x=0\) to \(x=2\). To find a definite integral:

• First, determine the antiderivative—a function whose derivative is the integrand (the function being integrated).
• Then, apply the Fundamental Theorem of Calculus, which allows us to evaluate the integral at specific upper and lower limits.

Evaluating definite integrals helps in various fields, such as finding the total displacement given a velocity function or determining the total area under curves in physics, economics, and engineering.

Antiderivative

The term "antiderivative" plays a crucial role when solving problems related to integration, especially with definite integrals. An antiderivative of a function \(f(x)\) is another function \(F(x)\) such that \(F'(x) = f(x)\). In simpler terms, it is the reverse process of differentiation. To find the antiderivative of \(3x^2 + 2x\), use basic integration rules:

• The Power Rule: For any term \(x^n\), the antiderivative is \( \frac{x^{n+1}}{n+1} \).
• Apply it to each term separately: for \(3x^2\), the antiderivative is \( x^3 \), and for \(2x\), it is \( x^2 \).
• Combine these results to form the complete antiderivative function: \( x^3 + x^2 + C \), where \(C\) is an arbitrary constant.

Understanding antiderivatives is essential as they form the basis for calculating definite integrals and applying the Fundamental Theorem of Calculus.

Integration by Parts

Integration by parts is a technique derived from the product rule for differentiation and is used when integrating products of functions that are not directly integrable. Although it was not directly applied in this exercise, understanding it as part of integral calculus is useful. The formula for integration by parts is: \[ \int u \, dv = uv - \int v \, du \] Where:

• \(u\) and \(dv\) are parts of the integrand \(f(x)\).
• \(du\) is the derivative of \(u\), and \(v\) is the antiderivative of \(dv\).

In practice, you choose \(u\) and \(dv\) to simplify the integral on the right. This method is particularly handy for integrals involving logarithmic, exponential, and trigonometric functions. Even though the integral \( \int_{0}^{2}(3x^2 + 2x) \, dx \) was solved through straightforward antiderivatives, familiarity with various methods provides a comprehensive understanding of integration challenges.