Math / Calculus

Questionstion 5
Consider $\int \frac{x^{2}+1}{x^{3}+3 x} d x$ What is the most efficient u-substitution? Select only ONE answer. $u=x^{3}+3 x$ $\mathrm{u}=\mathrm{x}+1$ $\mathrm{u}=\mathrm{x}$ $u=x^{3}$ $y=x^{2}+1$ $y=x^{2}$

Studdy Solution

STEP 1

What is this asking? Which substitution makes this integral, $\int \frac{x^{2}+1}{x^{3}+3x} dx$ , the easiest to solve? Watch out! Don't just pick a substitution randomly!
Think about how the substitution will affect the whole integral, both the numerator and the denominator.

STEP 2

1. Analyze the Integral
2. Explore the Options
3. Choose the Best Substitution

STEP 3

Alright, let's look at what we've got!
We have this integral $\int \frac{x^{2}+1}{x^{3}+3x} dx$ .
Notice that the denominator is $x^{3} + 3x$ .

STEP 4

Now, think about what happens when we take the derivative of $x^{3} + 3x$ .
We get $3x^{2} + 3$ , using the power rule!

STEP 5

Hmm, $3x^{2} + 3$ is pretty close to the numerator, $x^{2} + 1$ , isn't it?
In fact, $3x^{2} + 3$ is just $3$ times $x^{2} + 1$ .
This is a big clue!

STEP 6

Let's try the suggested substitution $u = x^{3} + 3x$ .
If we take the derivative of both sides with respect to $x$ , we get $\frac{du}{dx} = 3x^{2} + 3$ .

STEP 7

We can rewrite this as $du = (3x^{2} + 3) dx$ .
Now, notice that we can divide both sides by 3 to get $\frac{1}{3} du = (x^{2} + 1) dx$ .

STEP 8

Look at that!
The expression $(x^{2} + 1) dx$ shows up in our original integral.
This means we can replace it with $\frac{1}{3} du$ .
This is a great sign!

STEP 9

With the substitution $u = x^{3} + 3x$ , our integral becomes $\int \frac{1}{u} \cdot \frac{1}{3} du$ , which simplifies to $\frac{1}{3} \int \frac{1}{u} du$ .
Much easier, right?

STEP 10

The other substitutions won't simplify the integral as nicely.
For example, if we try $u = x$ , we get $du = dx$ , and the integral becomes $\int \frac{u^{2}+1}{u^{3}+3u} du$ , which isn't any simpler!

STEP 11

Since $u = x^{3} + 3x$ transforms the integral into a much simpler form, it's the most efficient substitution.
It lets us rewrite the integral in terms of $u$ and $du$ , making it much easier to solve.

STEP 12

The most efficient u-substitution is $u = x^{3} + 3x$ .

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Questionstion 5
Consider $\int \frac{x^{2}+1}{x^{3}+3 x} d x$ What is the most efficient u-substitution? Select only ONE answer. $u=x^{3}+3 x$ $\mathrm{u}=\mathrm{x}+1$ $\mathrm{u}=\mathrm{x}$ $u=x^{3}$ $y=x^{2}+1$ $y=x^{2}$

Studdy Solution

STEP 1

What is this asking? Which substitution makes this integral, $\int \frac{x^{2}+1}{x^{3}+3x} dx$ , the easiest to solve? Watch out! Don't just pick a substitution randomly!
Think about how the substitution will affect the whole integral, both the numerator and the denominator.

STEP 2

1. Analyze the Integral
2. Explore the Options
3. Choose the Best Substitution

STEP 3

Alright, let's look at what we've got!
We have this integral $\int \frac{x^{2}+1}{x^{3}+3x} dx$ .
Notice that the denominator is $x^{3} + 3x$ .

STEP 4

Now, think about what happens when we take the derivative of $x^{3} + 3x$ .
We get $3x^{2} + 3$ , using the power rule!

STEP 5

Hmm, $3x^{2} + 3$ is pretty close to the numerator, $x^{2} + 1$ , isn't it?
In fact, $3x^{2} + 3$ is just $3$ times $x^{2} + 1$ .
This is a big clue!

STEP 6

Let's try the suggested substitution $u = x^{3} + 3x$ .
If we take the derivative of both sides with respect to $x$ , we get $\frac{du}{dx} = 3x^{2} + 3$ .

STEP 7

We can rewrite this as $du = (3x^{2} + 3) dx$ .
Now, notice that we can divide both sides by 3 to get $\frac{1}{3} du = (x^{2} + 1) dx$ .

STEP 8

Look at that!
The expression $(x^{2} + 1) dx$ shows up in our original integral.
This means we can replace it with $\frac{1}{3} du$ .
This is a great sign!

STEP 9

With the substitution $u = x^{3} + 3x$ , our integral becomes $\int \frac{1}{u} \cdot \frac{1}{3} du$ , which simplifies to $\frac{1}{3} \int \frac{1}{u} du$ .
Much easier, right?

STEP 10

The other substitutions won't simplify the integral as nicely.
For example, if we try $u = x$ , we get $du = dx$ , and the integral becomes $\int \frac{u^{2}+1}{u^{3}+3u} du$ , which isn't any simpler!

STEP 11

Since $u = x^{3} + 3x$ transforms the integral into a much simpler form, it's the most efficient substitution.
It lets us rewrite the integral in terms of $u$ and $du$ , making it much easier to solve.

STEP 12

The most efficient u-substitution is $u = x^{3} + 3x$ .

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Questionstion 5Consider ∫x2+1x3+3xdx\int \frac{x^{2}+1}{x^{3}+3 x} d x∫x3+3xx2+1​dx What is the most efficient u-substitution? Select only ONE answer. u=x3+3xu=x^{3}+3 xu=x3+3x u=x+1\mathrm{u}=\mathrm{x}+1u=x+1 u=x\mathrm{u}=\mathrm{x}u=x u=x3u=x^{3}u=x3 y=x2+1y=x^{2}+1y=x2+1 y=x2y=x^{2}y=x2

Studdy Solution

STEP 1

STEP 2

1. Analyze the Integral2. Explore the Options3. Choose the Best Substitution

STEP 3

Alright, let's look at what we've got!We have this integral ∫x2+1x3+3xdx\int \frac{x^{2}+1}{x^{3}+3x} dx∫x3+3xx2+1​dx.Notice that the denominator is x3+3xx^{3} + 3xx3+3x.

STEP 4

Now, think about what happens when we take the derivative of x3+3xx^{3} + 3xx3+3x.We get 3x2+33x^{2} + 33x2+3, using the power rule!

STEP 5

Hmm, 3x2+33x^{2} + 33x2+3 is pretty close to the numerator, x2+1x^{2} + 1x2+1, isn't it?In fact, 3x2+33x^{2} + 33x2+3 is just 333 times x2+1x^{2} + 1x2+1.This is a big clue!

STEP 6

Let's try the suggested substitution u=x3+3xu = x^{3} + 3xu=x3+3x.If we take the derivative of both sides with respect to xxx, we get dudx=3x2+3\frac{du}{dx} = 3x^{2} + 3dxdu​=3x2+3.

STEP 7

We can rewrite this as du=(3x2+3)dxdu = (3x^{2} + 3) dxdu=(3x2+3)dx.Now, notice that we can divide both sides by 3 to get 13du=(x2+1)dx\frac{1}{3} du = (x^{2} + 1) dx31​du=(x2+1)dx.

STEP 8

Look at that!The expression (x2+1)dx(x^{2} + 1) dx(x2+1)dx shows up in our original integral.This means we can replace it with 13du\frac{1}{3} du31​du.This is a great sign!

STEP 9

With the substitution u=x3+3xu = x^{3} + 3xu=x3+3x, our integral becomes ∫1u⋅13du\int \frac{1}{u} \cdot \frac{1}{3} du∫u1​⋅31​du, which simplifies to 13∫1udu\frac{1}{3} \int \frac{1}{u} du31​∫u1​du.Much easier, right?

STEP 10

The other substitutions won't simplify the integral as nicely.For example, if we try u=xu = xu=x, we get du=dxdu = dxdu=dx, and the integral becomes ∫u2+1u3+3udu\int \frac{u^{2}+1}{u^{3}+3u} du∫u3+3uu2+1​du, which isn't any simpler!

STEP 11

Since u=x3+3xu = x^{3} + 3xu=x3+3x transforms the integral into a much simpler form, it's the most efficient substitution.It lets us rewrite the integral in terms of uuu and dududu, making it much easier to solve.

STEP 12

The most efficient u-substitution is u=x3+3xu = x^{3} + 3xu=x3+3x.

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Start learning now

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Questionstion 5
Consider $\int \frac{x^{2}+1}{x^{3}+3 x} d x$ What is the most efficient u-substitution? Select only ONE answer. $u=x^{3}+3 x$ $\mathrm{u}=\mathrm{x}+1$ $\mathrm{u}=\mathrm{x}$ $u=x^{3}$ $y=x^{2}+1$ $y=x^{2}$

1. Analyze the Integral
2. Explore the Options
3. Choose the Best Substitution

Alright, let's look at what we've got!
We have this integral $\int \frac{x^{2}+1}{x^{3}+3x} dx$ .
Notice that the denominator is $x^{3} + 3x$ .

Now, think about what happens when we take the derivative of $x^{3} + 3x$ .
We get $3x^{2} + 3$ , using the power rule!

Hmm, $3x^{2} + 3$ is pretty close to the numerator, $x^{2} + 1$ , isn't it?
In fact, $3x^{2} + 3$ is just $3$ times $x^{2} + 1$ .
This is a big clue!

Let's try the suggested substitution $u = x^{3} + 3x$ .
If we take the derivative of both sides with respect to $x$ , we get $\frac{du}{dx} = 3x^{2} + 3$ .

We can rewrite this as $du = (3x^{2} + 3) dx$ .
Now, notice that we can divide both sides by 3 to get $\frac{1}{3} du = (x^{2} + 1) dx$ .

Look at that!
The expression $(x^{2} + 1) dx$ shows up in our original integral.
This means we can replace it with $\frac{1}{3} du$ .
This is a great sign!

With the substitution $u = x^{3} + 3x$ , our integral becomes $\int \frac{1}{u} \cdot \frac{1}{3} du$ , which simplifies to $\frac{1}{3} \int \frac{1}{u} du$ .
Much easier, right?

The other substitutions won't simplify the integral as nicely.
For example, if we try $u = x$ , we get $du = dx$ , and the integral becomes $\int \frac{u^{2}+1}{u^{3}+3u} du$ , which isn't any simpler!

Since $u = x^{3} + 3x$ transforms the integral into a much simpler form, it's the most efficient substitution.
It lets us rewrite the integral in terms of $u$ and $du$ , making it much easier to solve.

The most efficient u-substitution is $u = x^{3} + 3x$ .