Solve Integral (3 + X^7)^3 X^6 Dx: Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of calculus to tackle a juicy integral problem. We'll break down the process step-by-step, making sure you understand not just the how, but also the why behind each move. So, let's get started!

The Integral Challenge

Our mission, should we choose to accept it (and we do!), is to evaluate the following integral:

$$\int (3 + x^7)^3 x^6 dx$$

This might look a bit intimidating at first glance, but don't worry! We've got a secret weapon in our mathematical arsenal: the u-substitution technique. This powerful tool allows us to simplify complex integrals by cleverly changing the variable of integration. Let's see how it works.

Unmasking the U-Substitution Method

The core idea behind u-substitution is to identify a part of the integrand (the function inside the integral) whose derivative also appears in the integral. This allows us to make a substitution that simplifies the expression and makes it easier to integrate.

In our case, we notice that the expression $(3 + x^7)$ is raised to the power of 3. The derivative of $x^7$ is $7x^6$, and we have an $x^6$ term sitting right there in the integral! This is a strong hint that u-substitution might be the way to go. So, let's define our 'u':

$$u = 3 + x^7$$

Now, we need to find the derivative of $u$ with respect to $x$, which we denote as $\frac{du}{dx}$:

$$\frac{du}{dx} = 7x^6$$

To make our substitution work, we need to express $dx$ in terms of $du$. We can do this by rearranging the equation above:

$$du = 7x^6 dx$$

$$dx = \frac{du}{7x^6}$$

The Substitution Tango

Now comes the fun part: substituting $u$ and $dx$ into our original integral. We replace $(3 + x^7)$ with $u$ and $dx$ with $\frac{du}{7x^6}$:

$$\int (3 + x^7)^3 x^6 dx = \int u^3 x^6 \frac{du}{7x^6}$$

Notice something beautiful? The $x^6$ terms cancel out! This is the magic of u-substitution in action. We're left with a much simpler integral:

$$\int u^3 x^6 \frac{du}{7x^6} = \int \frac{1}{7} u^3 du$$

We can pull the constant $\frac{1}{7}$ outside the integral:

$$\int \frac{1}{7} u^3 du = \frac{1}{7} \int u^3 du$$

Integrating the Simplified Expression

Now we have a straightforward integral to solve. We use the power rule for integration, which states that:

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$

where $C$ is the constant of integration. Applying this to our integral, we get:

$$\frac{1}{7} \int u^3 du = \frac{1}{7} \cdot \frac{u^4}{4} + C$$

$$\frac{1}{7} \cdot \frac{u^4}{4} + C = \frac{u^4}{28} + C$$

Back to the Original Variable

We're not quite done yet! We've expressed our result in terms of $u$, but we need to go back to our original variable, $x$. Remember that we defined $u = 3 + x^7$. So, we substitute this back into our expression:

$$\frac{u^4}{28} + C = \frac{(3 + x^7)^4}{28} + C$$

And there you have it! We've successfully evaluated the integral.

The Grand Finale: Our Solution

The final answer to our integral challenge is:

$$\int (3 + x^7)^3 x^6 dx = \frac{(3 + x^7)^4}{28} + C$$

Key Takeaways and Why This Matters

• U-Substitution is Your Friend: This technique is crucial for simplifying integrals where you can identify a function and its derivative within the integrand. Master this, and you'll conquer a huge chunk of integration problems.
• Spotting the Pattern: The key to u-substitution is recognizing the pattern of a function and its derivative. Practice makes perfect! The more you solve these problems, the quicker you'll become at identifying these patterns.
• The Power Rule is Fundamental: Remember the power rule for integration (and differentiation!). It's the bread and butter of calculus.
• Don't Forget the Constant of Integration: Always add that '+ C'! It represents the family of functions that have the same derivative.

Digging Deeper: Expanding Your Knowledge

Now that we've conquered this integral, let's explore some related concepts and ways to expand your calculus skills.

Practice Makes Perfect: More Integral Challenges

The best way to solidify your understanding is to practice! Try tackling similar integrals that require u-substitution. Here are a few ideas:

• $\int x^2 \sqrt{x^3 + 1} dx$
• $\int \sin(x) \cos(x) dx$
• $\int \frac{x}{x^2 + 1} dx$

Work through these, and you'll become a u-substitution master in no time!

Beyond U-Substitution: Other Integration Techniques

U-substitution is just one tool in the integration toolbox. As you delve deeper into calculus, you'll encounter other powerful techniques, such as:

• Integration by Parts: Used for integrals involving products of functions.
• Trigonometric Substitution: Useful for integrals containing square roots of quadratic expressions.
• Partial Fraction Decomposition: Used for integrating rational functions (fractions of polynomials).

Learning these techniques will significantly expand your ability to solve a wider range of integrals.

The Real-World Applications of Integration

Calculus isn't just an abstract mathematical concept; it has countless applications in the real world! Integration, in particular, is used in fields like:

• Physics: Calculating displacement, velocity, and acceleration.
• Engineering: Designing structures, analyzing circuits, and modeling fluid flow.
• Economics: Calculating consumer surplus and producer surplus.
• Statistics: Finding probabilities and expected values.

Understanding integration opens doors to a deeper understanding of the world around us.

Resources for Further Learning

If you're eager to learn more about integration and calculus, there are tons of fantastic resources available:

• Textbooks: Look for standard calculus textbooks like "Calculus" by James Stewart or "Calculus" by Thomas Finney.
• Online Courses: Platforms like Khan Academy, Coursera, and edX offer excellent calculus courses.
• Websites: Websites like Paul's Online Math Notes and Mathworld provide comprehensive explanations and examples.
• Practice Problems: Seek out practice problems online or in textbooks to test your skills.

A Final Word of Encouragement

Calculus can be challenging, but it's also incredibly rewarding. Don't be afraid to ask questions, seek help when you need it, and most importantly, keep practicing! With dedication and perseverance, you'll master the art of integration and unlock the power of calculus.

So, keep those integrals coming, guys! You've got this!