Math / Calculus

Question $\begin{array}{c} \int\left(x^{7}+x^{5}-x^{4}\right)^{6}\left(7 x^{6}+5 x^{4}-4 x^{3}\right) d x \\ \int\left(x^{7}+x^{5}-x^{4}\right)^{6}\left(7 x^{6}+5 x^{4}-4 x^{3}\right) d x= \end{array}$ $\square$ (Type an exact answer. Use parentheses to clearly denote the argument

Studdy Solution

STEP 1

What is this asking? We need to find the integral of a polynomial expression that looks like a function inside another function, raised to a power, multiplied by something that resembles the derivative of the inside function. Watch out! Don't forget to add the constant of integration because we're dealing with an indefinite integral!

STEP 2

1. Set up the substitution
2. Compute the derivative of $u$
3. Rewrite the integral in terms of $u$
4. Integrate with respect to $u$
5. Substitute back for $x$

STEP 3

Let's define our inside function as $u$ .
This is the heart of what we're doing!
We set $u = x^7 + x^5 - x^4$ .
This helps simplify the integral and makes it much easier to work with.

STEP 4

Now, we differentiate $u$ with respect to $x$ .
This gives us $\frac{du}{dx} = 7x^6 + 5x^4 - 4x^3$ .
This step is crucial because it helps us replace the $dx$ in our original integral with $du$ .

STEP 5

We can rewrite this as $du = (7x^6 + 5x^4 - 4x^3)dx$ .
Look closely!
This expression appears in our original integral.
This is a great sign, it means our substitution is working!

STEP 6

Using our substitution, we can transform the original integral into something much simpler: $\int (u)^6 du$ .
See how much cleaner that looks?

STEP 7

Now, we integrate with respect to $u$ .
Remember the power rule for integration: $\int u^n du = \frac{u^{n+1}}{n+1} + C$ , where $n \neq -1$ .

STEP 8

Applying the power rule, we get $\frac{u^{6+1}}{6+1} + C = \frac{u^7}{7} + C$ .
Don't forget that constant of integration, $C$ !
It's important to include it because the derivative of a constant is zero, so any constant could have been present in the original function.

STEP 9

Finally, we substitute the original expression for $u$ back into our result: $\frac{(x^7 + x^5 - x^4)^7}{7} + C$ .
This is our final answer in terms of $x$ .

STEP 10

$\frac{(x^7 + x^5 - x^4)^7}{7} + C$

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Question $\begin{array}{c} \int\left(x^{7}+x^{5}-x^{4}\right)^{6}\left(7 x^{6}+5 x^{4}-4 x^{3}\right) d x \\ \int\left(x^{7}+x^{5}-x^{4}\right)^{6}\left(7 x^{6}+5 x^{4}-4 x^{3}\right) d x= \end{array}$ $\square$ (Type an exact answer. Use parentheses to clearly denote the argument

Studdy Solution

STEP 1

STEP 2

1. Set up the substitution
2. Compute the derivative of $u$
3. Rewrite the integral in terms of $u$
4. Integrate with respect to $u$
5. Substitute back for $x$

STEP 3

Let's define our inside function as $u$ .
This is the heart of what we're doing!
We set $u = x^7 + x^5 - x^4$ .
This helps simplify the integral and makes it much easier to work with.

STEP 4

Now, we differentiate $u$ with respect to $x$ .
This gives us $\frac{du}{dx} = 7x^6 + 5x^4 - 4x^3$ .
This step is crucial because it helps us replace the $dx$ in our original integral with $du$ .

STEP 5

We can rewrite this as $du = (7x^6 + 5x^4 - 4x^3)dx$ .
Look closely!
This expression appears in our original integral.
This is a great sign, it means our substitution is working!

STEP 6

Using our substitution, we can transform the original integral into something much simpler: $\int (u)^6 du$ .
See how much cleaner that looks?

STEP 7

Now, we integrate with respect to $u$ .
Remember the power rule for integration: $\int u^n du = \frac{u^{n+1}}{n+1} + C$ , where $n \neq -1$ .

STEP 8

Applying the power rule, we get $\frac{u^{6+1}}{6+1} + C = \frac{u^7}{7} + C$ .
Don't forget that constant of integration, $C$ !
It's important to include it because the derivative of a constant is zero, so any constant could have been present in the original function.

STEP 9

Finally, we substitute the original expression for $u$ back into our result: $\frac{(x^7 + x^5 - x^4)^7}{7} + C$ .
This is our final answer in terms of $x$ .

STEP 10

$\frac{(x^7 + x^5 - x^4)^7}{7} + C$

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Studdy Solution

STEP 1

STEP 2

1. Set up the substitution2. Compute the derivative of uuu3. Rewrite the integral in terms of uuu4. Integrate with respect to uuu5. Substitute back for xxx

STEP 3

Let's **define** our inside function as uuu.This is the heart of what we're doing!We set u=x7+x5−x4u = x^7 + x^5 - x^4u=x7+x5−x4.This helps simplify the integral and makes it much easier to work with.

STEP 4

Now, we **differentiate** uuu with respect to xxx.This gives us dudx=7x6+5x4−4x3\frac{du}{dx} = 7x^6 + 5x^4 - 4x^3dxdu​=7x6+5x4−4x3.This step is crucial because it helps us replace the dxdxdx in our original integral with dududu.

STEP 5

We can **rewrite** this as du=(7x6+5x4−4x3)dxdu = (7x^6 + 5x^4 - 4x^3)dxdu=(7x6+5x4−4x3)dx.Look closely!This expression appears in our original integral.This is a great sign, it means our substitution is working!

STEP 6

Using our substitution, we can **transform** the original integral into something much simpler: ∫(u)6du\int (u)^6 du∫(u)6du.See how much cleaner that looks?

STEP 7

Now, we **integrate** with respect to uuu.Remember the power rule for integration: ∫undu=un+1n+1+C\int u^n du = \frac{u^{n+1}}{n+1} + C∫undu=n+1un+1​+C, where n≠−1n \neq -1n=−1.

STEP 8

Applying the power rule, we get u6+16+1+C=u77+C\frac{u^{6+1}}{6+1} + C = \frac{u^7}{7} + C6+1u6+1​+C=7u7​+C.Don't forget that constant of integration, CCC!It's important to include it because the derivative of a constant is zero, so any constant could have been present in the original function.

STEP 9

Finally, we **substitute** the original expression for uuu back into our result: (x7+x5−x4)77+C\frac{(x^7 + x^5 - x^4)^7}{7} + C7(x7+x5−x4)7​+C.This is our **final answer** in terms of xxx.

STEP 10

(x7+x5−x4)77+C\frac{(x^7 + x^5 - x^4)^7}{7} + C7(x7+x5−x4)7​+C

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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.

1. Set up the substitution
2. Compute the derivative of $u$
3. Rewrite the integral in terms of $u$
4. Integrate with respect to $u$
5. Substitute back for $x$

Let's define our inside function as $u$ .
This is the heart of what we're doing!
We set $u = x^7 + x^5 - x^4$ .
This helps simplify the integral and makes it much easier to work with.

Now, we differentiate $u$ with respect to $x$ .
This gives us $\frac{du}{dx} = 7x^6 + 5x^4 - 4x^3$ .
This step is crucial because it helps us replace the $dx$ in our original integral with $du$ .

We can rewrite this as $du = (7x^6 + 5x^4 - 4x^3)dx$ .
Look closely!
This expression appears in our original integral.
This is a great sign, it means our substitution is working!

Using our substitution, we can transform the original integral into something much simpler: $\int (u)^6 du$ .
See how much cleaner that looks?

Now, we integrate with respect to $u$ .
Remember the power rule for integration: $\int u^n du = \frac{u^{n+1}}{n+1} + C$ , where $n \neq -1$ .

Applying the power rule, we get $\frac{u^{6+1}}{6+1} + C = \frac{u^7}{7} + C$ .
Don't forget that constant of integration, $C$ !
It's important to include it because the derivative of a constant is zero, so any constant could have been present in the original function.

Finally, we substitute the original expression for $u$ back into our result: $\frac{(x^7 + x^5 - x^4)^7}{7} + C$ .
This is our final answer in terms of $x$ .

$\frac{(x^7 + x^5 - x^4)^7}{7} + C$