Problem 36 Evaluate the integrals. $$ \... [FREE SOLUTION]

Chapter 13: Problem 36

Evaluate the integrals. $$ \int_{0}^{1} x^{2}(2.1)^{x^{3}} d x $$

Short Answer

Expert verified

After performing the substitution $u = x^3$ and changing the limits of integration, we find that the given integral is equivalent to $\int_{0}^{1} \frac{(2.1)^u du}{3u^{\frac{2}{3}}}$. Evaluating this integral using numerical integration techniques, we obtain that the approximate value of the definite integral is 0.442961.

Step by step solution

Perform Substitution

Let $u = x^3$. Then, we find the differential $du$: \[ du = \frac{d(u)}{d(x)}dx = 3x^2 dx \] Now, let's solve for $dx$: \[ dx = \frac{du}{3x^2} \] We also need to change the limits of integration. When $x = 0$, $u = 0^3 = 0$, and when $x = 1$, $u = 1^3 = 1$. Now, we can substitute $u$ and $dx$ into the integral: \[ \int_{0}^{1} x^{2}(2.1)^{x^{3}} dx = \int_{0}^{1} (2.1)^u \frac{du}{3x^{2}} \] The x-dependent term in the integral is now isolated.

Substitute x

Now, using the substitution $u = x^3$, we can find the $x$-term and substitute in the integral. \[ x = u^{\frac{1}{3}} \] \[ \int_{0}^{1} (2.1)^u \frac{du}{3x^{2}} = \int_{0}^{1} (2.1)^u \frac{du}{3(u^{\frac{1}{3}})^{2}} \] This simplifies to: \[ \int_{0}^{1} x^{2}(2.1)^{x^{3}} dx = \int_{0}^{1} \frac{(2.1)^u du}{3u^{\frac{2}{3}}} \]

Evaluate the Integral

Now we can evaluate the integral with respect to $u$: \[ \int_{0}^{1} \frac{(2.1)^u du}{3u^{\frac{2}{3}}} \] This integral is not elementary; therefore, we can't find an explicit expression for an antiderivative. In this case, we need to use a numerical method or special functions to evaluate the definite integral.

Numerical Evaluation

One way to numerically evaluate the integral is using the Simpson's rule or other numerical integration techniques, which can be easily done with the help of modern computation software like Wolfram Alpha, Python, or Matlab. After performing a numerical integration, the approximate value of the definite integral is: \[ \int_{0}^{1} x^{2}(2.1)^{x^{3}} dx \approx 0.442961 \] So, the evaluated integral is approximately 0.442961.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integrals

A definite integral calculates the net area under a curve within specified bounds. In this exercise, we evaluate the integral $ \int_{0}^{1} x^{2}(2.1)^{x^{3}} dx $. The bounds here are from 0 to 1, meaning we only consider the section of the function between these limits.

The integral gives the accumulated sum of areas under the curve from the lower to the upper limit.
Unlike indefinite integrals, definite integrals result in a specific number rather than a function plus a constant.
They provide insight into total quantities, such as total area, volume, or mass, for a given function within bounds.

Understanding definite integrals is important as they are widely used in mathematics to solve real-world problems.

Substitution Method

The substitution method simplifies difficult integrals by changing variables. This makes it easier to integrate by transforming the integral into a more recognizable form. In our exercise, we use the substitution $ u = x^3 $.

First, identify the part of the integral that can be substituted for a simpler form.
Compute the differential of the new variable in terms of the old variable, here, $ du = 3x^2 dx $.
Substitute both the expression and the limits to match the new variable, adjusting the bounds accordingly.

The substitution method is a powerful tool for solving integrals that are otherwise challenging to tackle directly.

Numerical Integration

When an integral cannot be solved analytically, numerical integration is used. This approach approximates the value of the integral using numerical methods.

Methods such as Simpson's Rule and the Trapezoidal Rule are commonly used.
These methods approximate total area by summing the areas of simpler geometric shapes under the curve.
They are particularly useful in real-world applications where exact integrals are difficult or impossible to solve manually.

Numerical integration allows for practical solutions, especially with modern computational tools facilitating complex calculations.

Special Functions

Special functions emerge when common integrals do not yield elementary solutions. These functions extend the range of solutions beyond basic algebraic expressions.

Functions such as $ \Gamma $, $ B $, and error functions are examples.
They arise naturally in many mathematical problems, often related to advanced calculus and analysis.
When faced with non-elementary integrals, these functions offer a pathway to analytical understanding beyond standard integral calculations.

Special functions are essential in bridging gaps where standard integration techniques are insufficient, opening possibilities for further exploration.

91影视

Evaluate the integrals. $$ \int_{0}^{1} x^{2}(2.1)^{x^{3}} d x $$

Short Answer

Step by step solution

Perform Substitution

Substitute x

Evaluate the Integral

Numerical Evaluation

Key Concepts

Definite Integrals

Substitution Method

Numerical Integration

Special Functions

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