Math / Calculus

Question $\int \cos ^{3}(4 x) \sin (4 x) d x$

Studdy Solution

STEP 1

What is this asking? We need to find the indefinite integral of $\cos^3(4x) \cdot \sin(4x)$ with respect to $x$ .
Basically, we're looking for a function whose derivative is $\cos^3(4x) \cdot \sin(4x)$ . Watch out! Don't forget the chain rule and remember to add the constant of integration since it's an indefinite integral!

STEP 2

1. U-Substitution
2. Integrate
3. Substitute Back

STEP 3

Let's define our u!
We'll let $u = \cos(4x)$ .
This is a strategic choice because the derivative of $\cos(4x)$ is related to $\sin(4x)$ , which is also in our integral.
This substitution will simplify things greatly!

STEP 4

Now, let's find $du$ .
Taking the derivative of $u$ with respect to $x$ , we get $\frac{du}{dx} = -4\sin(4x)$ .
Remember the chain rule!
Multiplying both sides by $dx$ gives us $du = -4\sin(4x) dx$ .

STEP 5

We want to replace $\sin(4x) dx$ in our original integral with $du$ .
So, let's solve for $\sin(4x) dx$ in terms of $du$ .
Dividing both sides of $du = -4\sin(4x) dx$ by $-4$ gives us $\sin(4x) dx = -\frac{1}{4} du$ .

STEP 6

Now, let's rewrite our integral using the u substitution.
Substituting $u = \cos(4x)$ and $\sin(4x) dx = -\frac{1}{4} du$ , our integral becomes $\int u^3 \cdot \left(-\frac{1}{4}\right) du.$ We can pull the constant $-\frac{1}{4}$ out front: $-\frac{1}{4} \int u^3 du.$

STEP 7

Time to integrate!
Using the power rule for integration, we add 1 to the exponent and divide by the new exponent: $-\frac{1}{4} \int u^3 du = -\frac{1}{4} \cdot \frac{u^4}{4} + C.$ This simplifies to $-\frac{1}{16}u^4 + C$ , where C is our constant of integration.

STEP 8

Almost there!
Let's substitute $\cos(4x)$ back in for $u$ to get our final answer in terms of $x$ : $-\frac{1}{16}(\cos(4x))^4 + C.$ This can also be written as $-\frac{1}{16}\cos^4(4x) + C$ .

STEP 9

Our final answer is $-\frac{1}{16}\cos^4(4x) + C$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

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

Parents Influencer program Contact Policy Terms