Math / Calculus

QuestionPart 2 of 3
Use the Substitution Formula, $\int_{a}^{b} f(g(x)) \cdot g^{\prime}(x) d x=\int_{g(a)}^{g(b)} f(u)$ du where $g(x)=u$ , to evaluate the following integral. $\int_{1}^{10} \frac{2 \ln x}{x} d x$

Studdy Solution

STEP 1

What is this asking? We need to use the substitution formula to solve a definite integral with a logarithmic function inside. Watch out! Don't forget to change the limits of integration when you substitute!
Also, make sure you choose a u that simplifies the integral.

STEP 2

1. Define u and du
2. Substitute and solve
3. Calculate the boundaries

STEP 3

Let's define our u!
We want something that simplifies our integral, and we see that $\frac{1}{x}$ is the derivative of $\ln x$ .
So, let's set $u = \ln x$ .

STEP 4

Now, let's find du.
Taking the derivative of both sides, we get $\frac{du}{dx} = \frac{1}{x}.$ Multiplying both sides by $dx$ gives us $du = \frac{1}{x} \cdot dx$ .
Perfect!

STEP 5

Let's rewrite our integral using our substitution: $\int_{1}^{10} \frac{2 \ln x}{x} dx = \int_{1}^{10} 2 \cdot \ln x \cdot \frac{1}{x} dx = \int_{1}^{10} 2 \cdot u \cdot du.$ See how much cleaner that looks?

STEP 6

Now, we can integrate with respect to u: $\int_{1}^{10} 2 \cdot u \cdot du = 2 \int_{1}^{10} u \cdot du = 2 \cdot \left[ \frac{u^2}{2} \right]_{1}^{10} = \left[ u^2 \right]_{1}^{10}.$ We multiplied by $\frac{1}{2}$ and then by $2$ , adding to one, to get the power of u from $1$ to $2$ .

STEP 7

Remember, our original limits of integration were for x, not u.
We need to convert them!
Since $u = \ln x$ , when $x = 1$ , $u = \ln 1 = 0$ .
And when $x = 10$ , $u = \ln 10$ .

STEP 8

Now, we can plug in our u boundaries: $\left[ u^2 \right]_{1}^{10} = \left[ u^2 \right]_{0}^{\ln 10} = (\ln 10)^2 - (0)^2 = (\ln 10)^2.$

STEP 9

Our final answer is $(\ln 10)^2$ !

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QuestionPart 2 of 3
Use the Substitution Formula, $\int_{a}^{b} f(g(x)) \cdot g^{\prime}(x) d x=\int_{g(a)}^{g(b)} f(u)$ du where $g(x)=u$ , to evaluate the following integral. $\int_{1}^{10} \frac{2 \ln x}{x} d x$

Studdy Solution

STEP 1

What is this asking? We need to use the substitution formula to solve a definite integral with a logarithmic function inside. Watch out! Don't forget to change the limits of integration when you substitute!
Also, make sure you choose a u that simplifies the integral.

STEP 2

1. Define u and du
2. Substitute and solve
3. Calculate the boundaries

STEP 3

Let's define our u!
We want something that simplifies our integral, and we see that $\frac{1}{x}$ is the derivative of $\ln x$ .
So, let's set $u = \ln x$ .

STEP 4

Now, let's find du.
Taking the derivative of both sides, we get $\frac{du}{dx} = \frac{1}{x}.$ Multiplying both sides by $dx$ gives us $du = \frac{1}{x} \cdot dx$ .
Perfect!

STEP 5

Let's rewrite our integral using our substitution: $\int_{1}^{10} \frac{2 \ln x}{x} dx = \int_{1}^{10} 2 \cdot \ln x \cdot \frac{1}{x} dx = \int_{1}^{10} 2 \cdot u \cdot du.$ See how much cleaner that looks?

STEP 6

Now, we can integrate with respect to u: $\int_{1}^{10} 2 \cdot u \cdot du = 2 \int_{1}^{10} u \cdot du = 2 \cdot \left[ \frac{u^2}{2} \right]_{1}^{10} = \left[ u^2 \right]_{1}^{10}.$ We multiplied by $\frac{1}{2}$ and then by $2$ , adding to one, to get the power of u from $1$ to $2$ .

STEP 7

Remember, our original limits of integration were for x, not u.
We need to convert them!
Since $u = \ln x$ , when $x = 1$ , $u = \ln 1 = 0$ .
And when $x = 10$ , $u = \ln 10$ .

STEP 8

Now, we can plug in our u boundaries: $\left[ u^2 \right]_{1}^{10} = \left[ u^2 \right]_{0}^{\ln 10} = (\ln 10)^2 - (0)^2 = (\ln 10)^2.$

STEP 9

Our final answer is $(\ln 10)^2$ !

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Studdy Solution

STEP 1

What is this asking? We need to use the substitution formula to solve a definite integral with a logarithmic function inside. Watch out! Don't forget to change the limits of integration when you substitute!Also, make sure you choose a *u* that simplifies the integral.

STEP 2

1. Define *u* and *du*2. Substitute and solve3. Calculate the boundaries

STEP 3

Let's **define** our *u*!We want something that simplifies our integral, and we see that 1x\frac{1}{x}x1​ is the derivative of ln⁡x\ln xlnx.So, let's set u=ln⁡xu = \ln xu=lnx.

STEP 4

Now, let's find *du*.Taking the derivative of both sides, we get dudx=1x.\frac{du}{dx} = \frac{1}{x}.dxdu​=x1​. Multiplying both sides by dxdxdx gives us du=1x⋅dxdu = \frac{1}{x} \cdot dxdu=x1​⋅dx.Perfect!

STEP 5

STEP 6

STEP 7

Remember, our original limits of integration were for *x*, not *u*.We need to **convert** them!Since u=ln⁡xu = \ln xu=lnx, when x=1x = 1x=1, u=ln⁡1=0u = \ln 1 = 0u=ln1=0.And when x=10x = 10x=10, u=ln⁡10u = \ln 10u=ln10.

STEP 8

Now, we can **plug in** our *u* boundaries: [u2]110=[u2]0ln⁡10=(ln⁡10)2−(0)2=(ln⁡10)2.\left[ u^2 \right]_{1}^{10} = \left[ u^2 \right]_{0}^{\ln 10} = (\ln 10)^2 - (0)^2 = (\ln 10)^2.[u2]110​=[u2]0ln10​=(ln10)2−(0)2=(ln10)2.

STEP 9

Our **final answer** is (ln⁡10)2(\ln 10)^2(ln10)2!

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to use the substitution formula to solve a definite integral with a logarithmic function inside. Watch out! Don't forget to change the limits of integration when you substitute!
Also, make sure you choose a u that simplifies the integral.

1. Define u and du
2. Substitute and solve
3. Calculate the boundaries

Let's define our u!
We want something that simplifies our integral, and we see that $\frac{1}{x}$ is the derivative of $\ln x$ .
So, let's set $u = \ln x$ .

Now, let's find du.
Taking the derivative of both sides, we get $\frac{du}{dx} = \frac{1}{x}.$ Multiplying both sides by $dx$ gives us $du = \frac{1}{x} \cdot dx$ .
Perfect!

Remember, our original limits of integration were for x, not u.
We need to convert them!
Since $u = \ln x$ , when $x = 1$ , $u = \ln 1 = 0$ .
And when $x = 10$ , $u = \ln 10$ .

Now, we can plug in our u boundaries: $\left[ u^2 \right]_{1}^{10} = \left[ u^2 \right]_{0}^{\ln 10} = (\ln 10)^2 - (0)^2 = (\ln 10)^2.$

Our final answer is $(\ln 10)^2$ !