Find the antiderivative of the following function.f(x)=5x4f\left(x\right)=5x^4f(x)=5x4

F ( x ) = x 5 + C F\left(x\right)=x^5+C F ( x ) = x 5 + C

Find the antiderivative of the following function. f ( x ) = 5 x 4 f\left(x\right)=5x^4 f ( x ) = 5 x 4

this problem, we're asked to find the antiderivative of the following function. Now the way that we can do this is by thinking about what exactly would we have had to take the derivative of to get this function right here, 5x to the 4th power. And so finding this antiderivative, which we'll call big f of x, what I can see is that we have this 5 out in front here, and notice that the power is one less than the 5 we have in front. It almost kinda looks like we went through some kind of power rule here. So what I'm going to do is take x here, and I'm gonna erase it to this number in the front we see, which is the 5th power. Notice how if we had this situation, if we use the derivative power rule here, we could take that 5, bring it to the front, reduce the power by 1, that would give us 5x to the 4th. So we can see that this actually is the antiderivative here. But don't forget, you have to add the constant c whenever you take the antiderivative. So this right here would be the solution to the problem, and that's how you can solve these situations where you're trying to find the antiderivative. Hope you found this video helpful.

Find the antiderivative of the following function. f ( x ) = 5 x 4 f\left(x\right)=5x^4 f ( x ) = 5 x 4

F ( x ) = x 5 + C F\left(x\right)=x^5+C F ( x ) = x 5 + C

F ( x ) = 25 x 5 + C F\left(x\right)=25x^5+C F ( x ) = 25 x 5 + C

F ( x ) = 5 x 5 + C F\left(x\right)=5x^5+C F ( x ) = 5 x 5 + C

F ( x ) = 5 4 x 5 + C F\left(x\right)=\frac54x^5+C F ( x ) = 4 5 x 5 + C

7. Antiderivatives & Indefinite Integrals

Antiderivatives

Business Calculus

7. Antiderivatives & Indefinite Integrals

Antiderivatives

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Multiple Choice

Find the antiderivative of the following function.
$f\left(x\right)=5x^4$

$F\left(x\right)=25x^5+C$

$F\left(x\right)=x^5+C$

$F\left(x\right)=5x^5+C$

$F\left(x\right)=\frac54x^5+C$

Verified step by step guidance

Step 1: Recall the general rule for finding the antiderivative of a power function. The antiderivative of a function of the form f(x) = ax^n is given by F(x) = (a/(n+1))x^(n+1) + C, where C is the constant of integration.

Step 2: Identify the components of the given function f(x) = 5x^4. Here, a = 5 and n = 4.

Step 3: Apply the antiderivative rule. Increase the exponent n by 1, so n+1 = 4+1 = 5. Then divide the coefficient a by the new exponent, giving (5/5)x^5.

Step 4: Simplify the expression. The coefficient simplifies to 1, so the antiderivative becomes F(x) = x^5 + C.

Step 5: Conclude that the antiderivative of f(x) = 5x^4 is F(x) = x^5 + C, where C is the constant of integration.

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