Find the antiderivative of the following function.f(x)=10x9f\left(x\right)=10x^9f(x)=10x9

F ( x ) = x 10 + C F\left(x\right)=x^{10}+C F ( x ) = x 10 + C

Find the antiderivative of the following function. f ( x ) = 10 x 9 f\left(x\right)=10x^9 f ( x ) = 10 x 9

this problem, we are asked to find the antiderivative of the following function that we're given right here. Now notice in this problem, we're given 10 x to the 9th power. And if we wanna find its antiderivative, we need to think something to ourselves. What exactly can we take the derivative of to give us 10 x to the 9th? Now something I notice about this function that we have is we have x raised to a power, and we have this constant 10 out in front. This kinda looks like we did the power rule, because we have the 10 right here, then we have the one less exponent, the number one less. So what would happen if I took x and raised it to the 10th power? Right? Well, doing this, if I were to take this derivative, that means I multiply the 10 by the front and reduce the power by 1. Doing that would give me 10 x to the 10 minus 1, which is 9. So this right here would be the derivative if I took it, so that means that the antiderivative would be x to the 10th plus the constant of integration c. And remember, whenever you're dealing with these problems, you could always take the derivative of the function you just got to check to see if it comes out to what you have to to what the function was before, and it would. If we took this derivative here, we would get 10 x to the 9th, and then the derivative of any constant would just be 0, which would give us this function. So this is the solution. So that is how you can solve these types of situations where you're dealing with antiderivatives. Hope you found this video helpful.

Find the antiderivative of the following function. f ( x ) = 10 x 9 f\left(x\right)=10x^9 f ( x ) = 10 x 9

F ( x ) = x 10 + C F\left(x\right)=x^{10}+C F ( x ) = x 10 + C

F ( x ) = 10 x 10 + C F\left(x\right)=10x^{10}+C F ( x ) = 10 x 10 + C

F ( x ) = 100 x 10 + C F\left(x\right)=100x^{10}+C F ( x ) = 100 x 10 + C

F ( x ) = 10 9 x 10 + C F\left(x\right)=\frac{10}{9}x^{10}+C F ( x ) = 9 10 x 10 + 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.
$f\left(x\right)=10x^9$

$F\left(x\right)=x^{10}+C$

$F\left(x\right)=10x^{10}+C$

$F\left(x\right)=100x^{10}+C$

$F\left(x\right)=\frac{10}{9}x^{10}+C$

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) = 10x^9. Here, a = 10 and n = 9.

Step 3: Apply the antiderivative formula. Increase the exponent n by 1, so n+1 = 9+1 = 10. Then divide the coefficient a by the new exponent n+1. This gives (10/10)x^10.

Step 4: Simplify the expression. The coefficient simplifies to 1, so the antiderivative becomes F(x) = x^10 + C.

Step 5: Conclude that the correct antiderivative is F(x) = x^10 + C, where C is the constant of integration.

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