Math / Calculus

QuestionEvaluate the integral by making the given substitution. (Use $C$ for the constant of integration.) $\int e^{-5 x} d x, \quad u=-5 x$

Studdy Solution

STEP 1

What is this asking? We need to solve an integral by substituting part of it with $u$ , which will make it easier to solve! Watch out! Don't forget to substitute back to $x$ at the end, and always remember the constant of integration, $C$ !

STEP 2

1. Substitute and adjust
2. Solve the integral
3. Substitute back

STEP 3

We're given the substitution $u = -5x$ .
This is our starting point!

STEP 4

Now, we need to find $du$ by taking the derivative of $u$ with respect to $x$ .
So, $\frac{du}{dx} = \frac{d}{dx}(-5x) = -5$ .
This means $du = -5 \, dx$ .

STEP 5

We want to replace $dx$ in our original integral, so let's solve for $dx$ in terms of $du$ .
Dividing both sides of $du = -5 \, dx$ by $-5$ gives us $dx = -\frac{1}{5} \, du$ .

STEP 6

Let's substitute $u$ and $dx$ into our original integral: $\int e^{-5x} \, dx = \int e^u \cdot \left( -\frac{1}{5} \right) \, du = -\frac{1}{5} \int e^u \, du$ Look how much simpler that integral looks now!

STEP 7

The integral of $e^u$ with respect to $u$ is simply $e^u$ .
So, we have: $-\frac{1}{5} \int e^u \, du = -\frac{1}{5} e^u + C$ Remember that constant of integration, $C$ !
It's super important!

STEP 8

Now, let's substitute back to get our answer in terms of $x$ .
Remember, we defined $u = -5x$ .
So: $-\frac{1}{5} e^u + C = -\frac{1}{5} e^{-5x} + C$ And there we have it!

STEP 9

$-\frac{1}{5} e^{-5x} + C$

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QuestionEvaluate the integral by making the given substitution. (Use $C$ for the constant of integration.) $\int e^{-5 x} d x, \quad u=-5 x$

Studdy Solution

STEP 1

What is this asking? We need to solve an integral by substituting part of it with $u$ , which will make it easier to solve! Watch out! Don't forget to substitute back to $x$ at the end, and always remember the constant of integration, $C$ !

STEP 2

1. Substitute and adjust
2. Solve the integral
3. Substitute back

STEP 3

We're given the substitution $u = -5x$ .
This is our starting point!

STEP 4

Now, we need to find $du$ by taking the derivative of $u$ with respect to $x$ .
So, $\frac{du}{dx} = \frac{d}{dx}(-5x) = -5$ .
This means $du = -5 \, dx$ .

STEP 5

We want to replace $dx$ in our original integral, so let's solve for $dx$ in terms of $du$ .
Dividing both sides of $du = -5 \, dx$ by $-5$ gives us $dx = -\frac{1}{5} \, du$ .

STEP 6

Let's substitute $u$ and $dx$ into our original integral: $\int e^{-5x} \, dx = \int e^u \cdot \left( -\frac{1}{5} \right) \, du = -\frac{1}{5} \int e^u \, du$ Look how much simpler that integral looks now!

STEP 7

The integral of $e^u$ with respect to $u$ is simply $e^u$ .
So, we have: $-\frac{1}{5} \int e^u \, du = -\frac{1}{5} e^u + C$ Remember that constant of integration, $C$ !
It's super important!

STEP 8

Now, let's substitute back to get our answer in terms of $x$ .
Remember, we defined $u = -5x$ .
So: $-\frac{1}{5} e^u + C = -\frac{1}{5} e^{-5x} + C$ And there we have it!

STEP 9

$-\frac{1}{5} e^{-5x} + C$

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QuestionEvaluate the integral by making the given substitution. (Use CCC for the constant of integration.) ∫e−5xdx,u=−5x\int e^{-5 x} d x, \quad u=-5 x∫e−5xdx,u=−5x

Studdy Solution

STEP 1

What is this asking? We need to solve an integral by substituting part of it with uuu, which will make it easier to solve! Watch out! Don't forget to substitute back to xxx at the end, and always remember the constant of integration, CCC!

STEP 2

1. Substitute and adjust2. Solve the integral3. Substitute back

STEP 3

We're given the substitution u=−5xu = -5xu=−5x.This is our starting point!

STEP 4

Now, we need to find dududu by taking the derivative of uuu with respect to xxx.So, dudx=ddx(−5x)=−5\frac{du}{dx} = \frac{d}{dx}(-5x) = -5dxdu​=dxd​(−5x)=−5.This means du=−5 dxdu = -5 \, dxdu=−5dx.

STEP 5

We want to replace dxdxdx in our original integral, so let's solve for dxdxdx in terms of dududu.Dividing both sides of du=−5 dxdu = -5 \, dxdu=−5dx by −5-5−5 gives us dx=−15 dudx = -\frac{1}{5} \, dudx=−51​du.

STEP 6

STEP 7

The integral of eue^ueu with respect to uuu is simply eue^ueu.So, we have: −15∫eu du=−15eu+C -\frac{1}{5} \int e^u \, du = -\frac{1}{5} e^u + C −51​∫eudu=−51​eu+C Remember that constant of integration, CCC!It's super important!

STEP 8

Now, let's substitute back to get our answer in terms of xxx.Remember, we defined u=−5xu = -5xu=−5x.So: −15eu+C=−15e−5x+C -\frac{1}{5} e^u + C = -\frac{1}{5} e^{-5x} + C −51​eu+C=−51​e−5x+C And there we have it!

STEP 9

−15e−5x+C -\frac{1}{5} e^{-5x} + C −51​e−5x+C

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QuestionEvaluate the integral by making the given substitution. (Use $C$ for the constant of integration.) $\int e^{-5 x} d x, \quad u=-5 x$

What is this asking? We need to solve an integral by substituting part of it with $u$ , which will make it easier to solve! Watch out! Don't forget to substitute back to $x$ at the end, and always remember the constant of integration, $C$ !

1. Substitute and adjust
2. Solve the integral
3. Substitute back

We're given the substitution $u = -5x$ .
This is our starting point!

Now, we need to find $du$ by taking the derivative of $u$ with respect to $x$ .
So, $\frac{du}{dx} = \frac{d}{dx}(-5x) = -5$ .
This means $du = -5 \, dx$ .

We want to replace $dx$ in our original integral, so let's solve for $dx$ in terms of $du$ .
Dividing both sides of $du = -5 \, dx$ by $-5$ gives us $dx = -\frac{1}{5} \, du$ .

The integral of $e^u$ with respect to $u$ is simply $e^u$ .
So, we have: $-\frac{1}{5} \int e^u \, du = -\frac{1}{5} e^u + C$ Remember that constant of integration, $C$ !
It's super important!

Now, let's substitute back to get our answer in terms of $x$ .
Remember, we defined $u = -5x$ .
So: $-\frac{1}{5} e^u + C = -\frac{1}{5} e^{-5x} + C$ And there we have it!

$-\frac{1}{5} e^{-5x} + C$