Math  /  Calculus

QuestionConsider 6x7(4x8+9)6dx\int 6 x^{7}\left(4 x^{8}+9\right)^{6} d x What is the most appropriate u-substitution? Select only ONE answer. u=(4x8+9)6u=\left(4 x^{8}+9\right)^{6} u=6x7u=6 x^{7} u=4x8+9u=4 x^{8}+9 u=xu=x

Studdy Solution

STEP 1

What is this asking? We need to figure out the best way to simplify this integral using *u*-substitution! Watch out! Don't pick the first thing that looks like it might work.
Think about which substitution will make the integral *easiest* to solve.

STEP 2

1. Analyze the Integral
2. Identify the *u*-substitution
3. Test the *u*-substitution

STEP 3

Alright, let's take a look at this integral: 6x7(4x8+9)6dx\int 6x^7(4x^8 + 9)^6 dx.
It looks a little intimidating at first, but don't worry, we'll break it down!

STEP 4

The key to *u*-substitution is to find a function and its derivative within the integral.
Notice that 6x76x^7 looks suspiciously like the derivative of 4x8+94x^8 + 9!

STEP 5

Let's try setting u=4x8+9u = 4x^8 + 9.
If this is a good substitution, then its derivative, dudx\frac{du}{dx}, should show up in the integral as well.

STEP 6

**Calculate the derivative:** dudx=ddx(4x8+9)=32x7\frac{du}{dx} = \frac{d}{dx}(4x^8 + 9) = 32x^7

STEP 7

Now, we can rewrite this as du=32x7dxdu = 32x^7 dx.
We almost have a perfect match with our integral!
We have 6x7dx6x^7 dx in the integral, and we calculated 32x7dx32x^7 dx.
We can fix this by **isolating** dxdx: dx=du32x7dx = \frac{du}{32x^7}

STEP 8

Let's put our *u* and *dx* back into the **original integral**: 6x7(u)6du32x7\int 6x^7(u)^6 \frac{du}{32x^7}

STEP 9

Look at that!
The x7x^7 terms **divide to one**, which simplifies things a lot: 6(u)6du32=632u6du=316u6du\int 6(u)^6 \frac{du}{32} = \int \frac{6}{32}u^6 du = \int \frac{3}{16}u^6 du

STEP 10

Now *that* is an integral we can easily solve!
This confirms that u=4x8+9u = 4x^8 + 9 is the perfect *u*-substitution!

STEP 11

The most appropriate *u*-substitution is u=4x8+9u = 4x^8 + 9.

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