Math / Calculus

QuestionConsider $\int \sqrt{8 x^{6}+48 x} \cdot\left(x^{5}+1\right) d:$ What is the most appropriate u-substitution? $8 x^{6}+48 x$ $x^{5}+1$ $u=\sqrt{x}$ $u=x$

Studdy Solution

STEP 1

What is this asking? We need to find the best u-substitution to simplify this scary-looking integral! Watch out! Don't get tricked by the complexity of the expression.
Sometimes the simplest substitutions are the most effective.

STEP 2

1. Analyze the integrand
2. Propose a u-substitution
3. Test the substitution

STEP 3

Let's take a closer look at what we're dealing with.
We have an integral with a square root term, $\sqrt{8x^6 + 48x}$ , multiplied by $(x^5 + 1)$ .
It looks complicated, but don't worry, we'll break it down!

STEP 4

Notice the expression inside the square root: $8x^6 + 48x$ .
We can factor out $8x$ to get $8x(x^5 + 6)$ .
This is a good sign!
See how it relates to the other term $(x^5 + 1)$ ?
They both have an $x^5$ term.
This similarity is a huge clue for our u-substitution.

STEP 5

Let's try $u = 8x^6 + 48x$ .
This seems like a good starting point because it's the expression inside the square root.

STEP 6

If $u = 8x^6 + 48x$ , then the derivative is $\frac{du}{dx} = 48x^5 + 48$ .
We can factor out $48$ to get $\frac{du}{dx} = 48(x^5 + 1)$ .
This means $du = 48(x^5 + 1) dx$ .

STEP 7

Now, we can rewrite our integral in terms of u.
We have $\sqrt{8x^6 + 48x}$ which is just $\sqrt{u}$ .
Also, we have $(x^5 + 1)dx$ which is $\frac{1}{48} du$ .

STEP 8

Putting it all together, our integral becomes $\int \sqrt{u} \cdot \frac{1}{48} du = \frac{1}{48} \int u^{\frac{1}{2}} du.$

STEP 9

This is much simpler to integrate!
We can use the power rule to integrate $u^{\frac{1}{2}}$ .
Remember, the power rule says $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , where C is the constant of integration.

STEP 10

Applying the power rule, we get $\frac{1}{48} \int u^{\frac{1}{2}} du = \frac{1}{48} \cdot \frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C = \frac{1}{48} \cdot \frac{u^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{1}{48} \cdot \frac{2}{3} u^{\frac{3}{2}} + C = \frac{1}{72} u^{\frac{3}{2}} + C.$

STEP 11

Finally, substitute back $u = 8x^6 + 48x$ to get our final answer: $\frac{1}{72} (8x^6 + 48x)^{\frac{3}{2}} + C.$

STEP 12

The most appropriate u-substitution is $u = 8x^6 + 48x$ .

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