Math / Calculus

Question $\int 2x \cdot 1(x^2 - 3)^5 dx$ by substitution. (It is recommended that you check your results by differentiation.) Use capital C for the free constant. Answer:

Studdy Solution

STEP 1

What is this asking? Find the integral of $2x(x^2 - 3)^5$ using u-substitution, and then double-check the answer by taking the derivative. Watch out! Don't forget to substitute back to the original variable, x, and always add the constant of integration, C, since we're dealing with an indefinite integral!

STEP 2

1. Substitute
2. Integrate
3. Back-substitute
4. Check

STEP 3

Let's define our substitution!
We'll set $u = x^2 - 3$ .
This is our key to unlocking this integral!

STEP 4

Now, we need to find $\frac{du}{dx}$ .
Taking the derivative of u with respect to x gives us $\frac{du}{dx} = 2x$ .

STEP 5

Next, we want to express dx in terms of du.
We can rearrange our equation to get $du = 2x \cdot dx$ .
Perfect!

STEP 6

Now, let's rewrite our original integral using our substitution: $\int 2x(x^2 - 3)^5 dx = \int u^5 du$ .
See how much simpler that looks?

STEP 7

Time to integrate!
The power rule of integration tells us that $\int u^n du = \frac{u^{n+1}}{n+1} + C$ , where n is not equal to -1.

STEP 8

Applying the power rule to our integral, we get $\int u^5 du = \frac{u^{5+1}}{5+1} + C = \frac{u^6}{6} + C$ .
We're halfway there!

STEP 9

Remember, we started with x, so we need to finish with x!
Let's substitute $u = x^2 - 3$ back into our result: $\frac{u^6}{6} + C = \frac{(x^2 - 3)^6}{6} + C$ .
This is our potential solution!

STEP 10

To verify our answer, let's differentiate it with respect to x.
We'll use the chain rule: $\frac{d}{dx} \left( \frac{(x^2 - 3)^6}{6} + C \right) = \frac{6(x^2 - 3)^5}{6} \cdot 2x + 0 = (x^2 - 3)^5 \cdot 2x$ , which simplifies to $2x(x^2 - 3)^5$ .

STEP 11

This matches our original integrand, so our integration is correct!

STEP 12

The solution to the integral $\int 2x(x^2 - 3)^5 dx$ is $\frac{(x^2 - 3)^6}{6} + C$ .

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Question $\int 2x \cdot 1(x^2 - 3)^5 dx$ by substitution. (It is recommended that you check your results by differentiation.) Use capital C for the free constant. Answer:

Studdy Solution

STEP 1

STEP 2

1. Substitute
2. Integrate
3. Back-substitute
4. Check

STEP 3

Let's define our substitution!
We'll set $u = x^2 - 3$ .
This is our key to unlocking this integral!

STEP 4

Now, we need to find $\frac{du}{dx}$ .
Taking the derivative of u with respect to x gives us $\frac{du}{dx} = 2x$ .

STEP 5

Next, we want to express dx in terms of du.
We can rearrange our equation to get $du = 2x \cdot dx$ .
Perfect!

STEP 6

Now, let's rewrite our original integral using our substitution: $\int 2x(x^2 - 3)^5 dx = \int u^5 du$ .
See how much simpler that looks?

STEP 7

Time to integrate!
The power rule of integration tells us that $\int u^n du = \frac{u^{n+1}}{n+1} + C$ , where n is not equal to -1.

STEP 8

Applying the power rule to our integral, we get $\int u^5 du = \frac{u^{5+1}}{5+1} + C = \frac{u^6}{6} + C$ .
We're halfway there!

STEP 9

Remember, we started with x, so we need to finish with x!
Let's substitute $u = x^2 - 3$ back into our result: $\frac{u^6}{6} + C = \frac{(x^2 - 3)^6}{6} + C$ .
This is our potential solution!

STEP 10

To verify our answer, let's differentiate it with respect to x.
We'll use the chain rule: $\frac{d}{dx} \left( \frac{(x^2 - 3)^6}{6} + C \right) = \frac{6(x^2 - 3)^5}{6} \cdot 2x + 0 = (x^2 - 3)^5 \cdot 2x$ , which simplifies to $2x(x^2 - 3)^5$ .

STEP 11

This matches our original integrand, so our integration is correct!

STEP 12

The solution to the integral $\int 2x(x^2 - 3)^5 dx$ is $\frac{(x^2 - 3)^6}{6} + C$ .

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Question∫2x⋅1(x2−3)5dx\int 2x \cdot 1(x^2 - 3)^5 dx∫2x⋅1(x2−3)5dx by substitution. (It is recommended that you check your results by differentiation.) Use capital C for the free constant. Answer:

Studdy Solution

STEP 1

STEP 2

1. Substitute2. Integrate3. Back-substitute4. Check

STEP 3

Let's **define** our substitution!We'll set u=x2−3u = x^2 - 3u=x2−3.This is our **key** to unlocking this integral!

STEP 4

Now, we need to find dudx\frac{du}{dx}dxdu​.Taking the derivative of *u* with respect to *x* gives us dudx=2x\frac{du}{dx} = 2xdxdu​=2x.

STEP 5

Next, we want to express *dx* in terms of *du*.We can rearrange our equation to get du=2x⋅dxdu = 2x \cdot dxdu=2x⋅dx.Perfect!

STEP 6

Now, let's **rewrite** our original integral using our substitution: ∫2x(x2−3)5dx=∫u5du\int 2x(x^2 - 3)^5 dx = \int u^5 du∫2x(x2−3)5dx=∫u5du.See how much simpler that looks?

STEP 7

Time to **integrate**!The power rule of integration tells us that ∫undu=un+1n+1+C\int u^n du = \frac{u^{n+1}}{n+1} + C∫undu=n+1un+1​+C, where *n* is not equal to -1.

STEP 8

Applying the power rule to our integral, we get ∫u5du=u5+15+1+C=u66+C\int u^5 du = \frac{u^{5+1}}{5+1} + C = \frac{u^6}{6} + C∫u5du=5+1u5+1​+C=6u6​+C.We're halfway there!

STEP 9

Remember, we started with *x*, so we need to finish with *x*!Let's **substitute** u=x2−3u = x^2 - 3u=x2−3 back into our result: u66+C=(x2−3)66+C\frac{u^6}{6} + C = \frac{(x^2 - 3)^6}{6} + C6u6​+C=6(x2−3)6​+C.This is our **potential solution**!

STEP 10

STEP 11

This matches our original integrand, so our integration is **correct**!

STEP 12

The solution to the integral ∫2x(x2−3)5dx\int 2x(x^2 - 3)^5 dx∫2x(x2−3)5dx is (x2−3)66+C\frac{(x^2 - 3)^6}{6} + C6(x2−3)6​+C.

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Question $\int 2x \cdot 1(x^2 - 3)^5 dx$ by substitution. (It is recommended that you check your results by differentiation.) Use capital C for the free constant. Answer:

1. Substitute
2. Integrate
3. Back-substitute
4. Check

Let's define our substitution!
We'll set $u = x^2 - 3$ .
This is our key to unlocking this integral!

Now, we need to find $\frac{du}{dx}$ .
Taking the derivative of u with respect to x gives us $\frac{du}{dx} = 2x$ .

Next, we want to express dx in terms of du.
We can rearrange our equation to get $du = 2x \cdot dx$ .
Perfect!

Now, let's rewrite our original integral using our substitution: $\int 2x(x^2 - 3)^5 dx = \int u^5 du$ .
See how much simpler that looks?

Time to integrate!
The power rule of integration tells us that $\int u^n du = \frac{u^{n+1}}{n+1} + C$ , where n is not equal to -1.

Applying the power rule to our integral, we get $\int u^5 du = \frac{u^{5+1}}{5+1} + C = \frac{u^6}{6} + C$ .
We're halfway there!

Remember, we started with x, so we need to finish with x!
Let's substitute $u = x^2 - 3$ back into our result: $\frac{u^6}{6} + C = \frac{(x^2 - 3)^6}{6} + C$ .
This is our potential solution!

This matches our original integrand, so our integration is correct!

The solution to the integral $\int 2x(x^2 - 3)^5 dx$ is $\frac{(x^2 - 3)^6}{6} + C$ .