Question

Evaluate the following integrals. Include absolute values only when needed. $$\int x^{2} 10^{x^{3}} d x$$

Short answer

Question: Evaluate the given integral $$\int x^2 10^{x^3} dx$$. Answer: The integral $$\int x^2 10^{x^3} dx$$ evaluates to $$\frac{1}{3\ln{10}} 10^{x^{3}}+C$$.

Step-by-step solution

Identify the appropriate substitution

We can see that $$x^2$$ is the derivative of $$x^3$$, so we can substitute $$u = x^3$$.

Compute the derivative of the substitution variable

Now, we need to find the derivative of $$u$$ with respect to $$x$$: $$\frac{d u}{d x}=3 x^{2}$$

Calculate the differential dx

Now, let's express $$d x$$ in terms of $$d u$$. $$d x=\frac{d u}{3 x^{2}}$$

Substitute the variables

Substitute $$x^3$$ with $$u$$ and $$d x$$ with the expression we found in terms of $$d u$$. $$\int x^{2} 10^{x^{3}} d x = \int 10^{u} \frac{1}{3}d u$$

Evaluate the integral

The integral now becomes easier to evaluate: $$\int 10^{u} \frac{1}{3}d u = \frac{1}{3}\int 10^{u}d u$$ To evaluate this integral, we use the formula $$\int a^{u}du=\frac{1}{\ln{a}}a^{u}+C$$ where $$a$$ is a constant and $$C$$ is the constant of integration. $$\frac{1}{3}\int 10^{u}d u =\frac{1}{3} \cdot \frac{1}{\ln{10}} 10^{u}+C$$

Replace the substitution variable with the original variable

Now, we replace the $$u$$ with $$x^3$$ to get the solution in terms of $$x$$: $$\frac{1}{3} \cdot \frac{1}{\ln{10}} 10^{u}+C = \frac{1}{3} \cdot \frac{1}{\ln{10}} 10^{x^3}+C$$ The final answer is: $$\int x^{2} 10^{x^{3}} d x = \frac{1}{3\ln{10}} 10^{x^{3}}+C$$

Key concepts

Substitution Method

The substitution method is a powerful tool in calculus for simplifying the process of integration. It transforms a complicated integral into a simpler one by changing the variable of integration. This technique often involves identifying a part of the integral that can be replaced by a new variable, typically called \( u \). In this way, the integration becomes manageable. Here's how the method works:

• Select the substitution: Scan through the integral to find a suitable expression, generally a portion of the function whose derivative is also present. For example, in the integral \( \int x^{2} 10^{x^{3}} d x \), notice that \( x^2 \) is the derivative of \( x^3 \).
• Differentiate the substitution variable: Compute \( \frac{d u}{d x} \) for the chosen substitution \( u = x^3 \). This gives us \( 3x^2 \).
• Express \( dx \) in terms of \( du \): Rearrange to find \( d x = \frac{d u}{3 x^{2}} \).
• Perform the substitution: Substitute \( u \) into the integral to transform its form. The original integral becomes \( \int 10^{u} \frac{1}{3}d u \), which is easier to solve.

This approach simplifies the computation, making the integration process much more straightforward.

Definite Integral

Definite integrals help us find the area under a curve or between curves and are crucial in understanding the accumulation of quantities. While the original exercise does not specify bounds, understanding definite integrals enriches the concept of integration.A definite integral is expressed as \( \int_{a}^{b} f(x) \, dx \). Here's a quick overview of its core ideas:

• Bounds of Integration: The values \( a \) and \( b \) are the limits between which integration is performed. These represent the interval on the x-axis over which you are finding the area or accumulation.
• Evaluating the Integral: Find the anti-derivative of the function, often denoted as \( F(x) \), and compute \( F(b) - F(a) \). This gives the net area.
• Applications: Used in physics, engineering, and economics to determine quantities like total distance, area, and resource consumption between two points.

Though our exercise results in an indefinite integration, understanding definite integrals primes students for real-world applications and problems.

Indefinite Integral

Indefinite integrals are fundamental to calculus, providing the general form of antiderivatives. These do not have specified limits and result in a function plus a constant. In the exercise, the integral \( \int x^{2} 10^{x^{3}} d x \) is an indefinite integral, showcasing the principles involved.The jerk behind indefinite integrals is as follows:

• Finding Antiderivatives: It revolves around determining a function whose derivative yields the integrand. In our example, the solution involves substituting and then integrating \( 10^u \), resulting in the form \( \frac{1}{3\ln{10}} 10^{x^3} + C \).
• Constant of Integration: Represented by \( C \), this constant accounts for all potential antiderivatives since the derivative of a constant is zero. This is why the solution includes \( C \).
• Purpose and Uses: Indefinite integrals enable us to determine families of functions and solve differential equations which model real systems.

Understanding indefinite integrals is crucial for solving general problems involving rates of change and cumulative quantities.