Find the following indefinite integral. ∫ 4 x 3 d x \int4x^3dx ∫ 4 x 3 d x

This problem, we are asked to find the following indefinite integral, and this is the integral that we have right here. Now remember, for the indefinite integral, what you're trying to do is find the antiderivative of this function, and that is going to give you your answer. And since I can see we have 4 x cubed right here, notice we're dealing with the power function. And I can see that we have this situation where the 4 is this number that's out in front, and then we have this 3, which is just 4 reduced by 1. So it looks like we have some kind of power rule that happened here when we initially took the derivative. So what I'm going to do is try x to the 4th power, because I know if I use the power rule for derivatives, that will give me 4x to the 3rd power, which is exactly what we have here. So x to the 4th plus c would be the solution. If we wanna check this to make sure it's accurate, well, let's just try taking the derivative of this function. So if I take the derivative of what I just found, x to the 4th plus c, well, I can use the power rule. Using the power rule on the x to the 4th, that's going to give me 4 x cubed. And then the derivative of c or any constant is just 0. So we end up with 4 x cubed, and notice how this is the same result that we had started with inside the integral. This is our integrand that we got, which means this here would be the solution to our problem. So this is how you can find these indefinite integrals, as well as check to make sure you did the indefinite integral correctly. See you in the next video.

Find the following indefinite integral. ∫ 4 x 3 d x \int4x^3dx ∫ 4 x 3 d x

4 3 x 4 + C \frac43x^4+C 3 4 x 4 + C

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Business Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

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Multiple Choice

Find the following indefinite integral.
$\int4x^3dx$

$x^4$

$x^4+C$

$\frac43x^4+C$

$\frac43x^4$

Verified step by step guidance

Step 1: Recall the power rule for integration, which states that for any term of the form ∫x^n dx, the integral is (x^(n+1))/(n+1) + C, where n ≠ -1.

Step 2: Identify the term to integrate in the given problem. Here, the term is 4x^3. The coefficient 4 is a constant and can be factored out of the integral.

Step 3: Apply the power rule to the term x^3. Increase the exponent by 1 (3 + 1 = 4) and divide by the new exponent (4). This gives (x^4)/4.

Step 4: Multiply the result by the constant 4 that was factored out earlier. This simplifies to (4/4)x^4, which reduces to x^4.

Step 5: Add the constant of integration, C, to account for the indefinite integral. The final expression is x^4 + C.

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Find the following indefinite integral.∫4x3dx\int4x^3dx∫4x3dx

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Indefinite Integrals practice set

Find the following indefinite integral.
$\int4x^3dx$