GoMim AI | Integral Calculator – Learn How to Solve Integrals Step by Step

Integrals are a fundamental concept in mathematics, introduced in calculus. They help calculate areas, volumes, and accumulated quantities, and are widely used in physics, engineering, and data analysis. Understanding integrals provides insight into how things change and accumulate over time.

What is an Integral?

An integral is a mathematical concept that represents the area under a curve in a graph, usually between two points. To put it simply, if you have a curve that represents a function, the integral helps you find out how much space is enclosed by that curve and the x-axis. The process of finding an integral is called integration. In mathematical notation, the integral of a function \( f(x) \) from \( a \) to \( b \) is written as: $$ \int_{a}^{b} f(x) \, dx $$ This expression asks for the total sum of the infinitesimally small rectangles under the curve \( f(x) \) from \( x = a \) to \( x = b \).

Why Are Integrals Important?

Integrals are essential in mathematics and its applications because they allow us to calculate areas, volumes, central points, and many other quantities. In calculus, they are used alongside derivatives to solve problems involving change and motion. For example, in physics, integrals help calculate quantities like work done by a force, or the center of mass of an object. In engineering, they can be used to determine the necessary materials or resources for a project. Additionally, integrals are crucial in data analysis, helping to model and predict trends based on accumulated data. They are often tested in exams and are a foundational skill for anyone studying advanced mathematics or related fields.

How to Calculate an Integral Step by Step

Calculating an integral involves the following steps:

1. Identify the function \( f(x) \) to be integrated and the interval \([a, b]\).
2. Find the antiderivative of \( f(x) \), which is a function \( F(x) \) such that \( F'(x) = f(x) \).
3. Evaluate the antiderivative at the endpoints \( a \) and \( b \).
4. Subtract the values: \( F(b) - F(a) \) to find the integral.

Example:

Calculate the integral of \( f(x) = x^2 \) from \( 0 \) to \( 3 \).

1. Identify \( f(x) = x^2 \), interval \([0, 3]\).
2. Antiderivative of \( x^2 \) is \( F(x) = \frac{x^3}{3} \).
3. Evaluate \( F(x) \) at \( 3 \) and \( 0 \):
4. \( F(3) = \frac{3^3}{3} = 9 \)
5. \( F(0) = \frac{0^3}{3} = 0 \)
6. Subtract: \( 9 - 0 = 9 \).

Thus, the integral of \( x^2 \) from \( 0 \) to \( 3 \) is \( 9 \).

How do I integrate the function x sin 2x step by step?

To integrate the function $x \sin(2x)$, you will use the method of Integration by Parts.

The standard formula for Integration by Parts is:

$$\int u \, dv = uv - \int v \, du$$

Here is the step-by-step solution:

Step 1: Choose $u$ and $dv$

Using the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to prioritize which term to differentiate, we assign $u$ to the algebraic term ($x$) and $dv$ to the trigonometric term ($\sin(2x)$).

• $u = x$
• $dv = \sin(2x) \, dx$

Step 2: Differentiate $u$ and Integrate $dv$

Next, we find the differential $du$ and the function $v$.

• Differentiate $u$:
• $$du = dx$$
• Integrate $dv$:
• $$v = \int \sin(2x) \, dx = -\frac{1}{2}\cos(2x)$$
• (Note: The integral of $\sin(kx)$ is $-\frac{1}{k}\cos(kx)$)

Step 3: Apply the Formula

Substitute your values for $u$, $v$, $du$, and $dv$ into the Integration by Parts formula:

$$\int x \sin(2x) \, dx = \underbrace{x}_{u} \underbrace{\left( -\frac{1}{2}\cos(2x) \right)}_{v} - \int \underbrace{\left( -\frac{1}{2}\cos(2x) \right)}_{v} \underbrace{dx}_{du}$$

Step 4: Simplify and Solve the Remaining Integral

First, clean up the constants to make the integral easier to solve:

$$= -\frac{1}{2}x \cos(2x) + \frac{1}{2} \int \cos(2x) \, dx$$

Now, evaluate the remaining integral $\int \cos(2x) \, dx$.

(Recall that the integral of $\cos(kx)$ is $\frac{1}{k}\sin(kx)$):

$$\int \cos(2x) \, dx = \frac{1}{2}\sin(2x)$$

Substitute this back into your equation:

$$= -\frac{1}{2}x \cos(2x) + \frac{1}{2} \left( \frac{1}{2}\sin(2x) \right) + C$$

Final Answer

Multiplying the fractions gives the final result:

$$\int x \sin(2x) \, dx = -\frac{1}{2}x \cos(2x) + \frac{1}{4}\sin(2x) + C$$

Related Practice Problem

Problem: Find the integral of \( f(x) = 2x \) from \( 1 \) to \( 4 \).

Step-by-step Solution:

1. Identify \( f(x) = 2x \), interval \([1, 4]\).

2. The antiderivative of \( 2x \) is \( F(x) = x^2 \).

3. Evaluate \( F(x) \) at \( 4 \) and \( 1 \):

- \( F(4) = 4^2 = 16 \)

- \( F(1) = 1^2 = 1 \)

4. Subtract: \( 16 - 1 = 15 \).

Thus, the integral of \( 2x \) from \( 1 \) to \( 4 \) is \( 15 \).

FAQ

Q: What is an integral in simple terms?

A: An integral is a mathematical tool used to calculate the area under a curve.

Q: What are the different types of integrals?

A: The two main types are definite and indefinite integrals. Definite integrals calculate the area under a curve between two points, while indefinite integrals find the general form of antiderivatives.

Q: How are integrals used in real life?

A: Integrals are used in various fields like physics for calculating work done, in engineering for material estimation, and in economics for finding consumer surplus.

Q: What is the difference between an integral and a derivative?

A: A derivative measures the rate of change of a function, while an integral measures the total accumulation of quantities.

Q: Can integrals be negative?

A: Yes, integrals can be negative if the function lies below the x-axis over the interval.

Q: How does an integral calculator work?

A: An integral calculator uses algorithms to find the antiderivative of a function and evaluate it at given limits to provide the integral value.

Conclusion

Understanding integrals is key to mastering calculus and applying mathematical concepts to real-world scenarios. Whether you're calculating areas, volumes, or other quantities, integrals are indispensable. Use AI tools like GoMim's Math Solver to assist in your learning and problem-solving processes, making the task easier and more efficient.