For the following function f(x)f\left(x\right)f(x), find the antiderivative F(x)F\left(x\right)F(x) that satisfies the given condition.f(x)=5x4f\left(x\right)=5x^4f(x)=5x4; F(0)=1F\left(0\right)=1F(0)=1

F ( x ) = x 5 + 1 F\left(x\right)=x^5+1 F ( x ) = x 5 + 1

For the following function f ( x ) f\left(x\right) f ( x ) , find the antiderivative F ( x ) F\left(x\right) F ( x ) that satisfies the given condition. f ( x ) = 5 x 4 f\left(x\right)=5x^4 f ( x ) = 5 x 4 ; F ( 0 ) = 1 F\left(0\right)=1 F ( 0 ) = 1

the following function below, f of x, find the antiderivative big f of x that satisfies the given condition. So we have this function given to us, and we're asked to find the antiderivative. Now the first step I'm going to do is by finding this antiderivative function to give me the general solution. In doing this, well, I need to think about what I need to take the derivative of to give me 5x to the 4th. And if I think about it, that's gonna be x to the 5th, because I know by using this power rule, the 5 is gonna come out to the front, and then reducing this power by 1 will give me that 4 there. So that's going to be the function for the antiderivative, but we always have to add plus our constant c. So this here is the general case, the general solution where we have this plus c at the end here. We were trying to find the antiderivative that satisfies this condition. So that means we need to figure out what the c value is. And to do that, well, I can go ahead and plug this point in. So we have this function now. So we can see that f of 0 right here is equal to 1. So we're gonna have 0 to the 5th power plus c plugging in 0 for x, and we know that f of 0 is 1. So that means that 1 is going to equal 0 to the 5th power which is 0 plus c, meaning that c is equal to 1. So going back up to this function, we're going to have that our function f of x is equal to x to the 5th power plus this c value of 1. And that's going to be the solution for our function, and that is how you can solve this problem. Hope you found this video helpful, and let's move on.

For the following function f ( x ) f\left(x\right) f ( x ) , find the antiderivative F ( x ) F\left(x\right) F ( x ) that satisfies the given condition. f ( x ) = 5 x 4 f\left(x\right)=5x^4 f ( x ) = 5 x 4 ; F ( 0 ) = 1 F\left(0\right)=1 F ( 0 ) = 1

F ( x ) = x 5 + 1 F\left(x\right)=x^5+1 F ( x ) = x 5 + 1

F ( x ) = 5 x 5 + 1 F\left(x\right)=5x^5+1 F ( x ) = 5 x 5 + 1

F ( x ) = 25 x 5 + 1 F\left(x\right)=25x^5+1 F ( x ) = 25 x 5 + 1

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

7. Antiderivatives & Indefinite Integrals

Antiderivatives

Calculus

7. Antiderivatives & Indefinite Integrals

Antiderivatives

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Multiple Choice

For the following function $f\left(x\right)$ , find the antiderivative $F\left(x\right)$ that satisfies the given condition.
$f\left(x\right)=5x^4$ ; $F\left(0\right)=1$

$F\left(x\right)=5x^5+1$

$F\left(x\right)=x^5+1$

$F\left(x\right)=25x^5+1$

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

Verified step by step guidance

To find the antiderivative F(x) of the function f(x) = 5x^4, we need to apply the power rule for integration. The power rule states that the antiderivative of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration.

Apply the power rule to f(x) = 5x^4. Increase the exponent by 1, which gives us x^5, and divide by the new exponent. This results in (5/5)x^5 = x^5.

Now, we have F(x) = x^5 + C, where C is the constant of integration. To find the value of C, use the condition F(0) = 1.

Substitute x = 0 into F(x) = x^5 + C, which gives us F(0) = 0^5 + C = C. Since F(0) = 1, we have C = 1.

Thus, the antiderivative that satisfies the given condition is F(x) = x^5 + 1.

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