Find the following indefinite integral. ∫ 100 x 99 d x \int100x^{99}dx ∫ 100 x 99 d x

this problem, we are asked to find the following indefinite integral, and we are given this function right here that we are integrating with respect to x. Now to solve this problem, what we wanna do is find what the antiderivative is of this integrand that shows up inside the integral. If we can find the antiderivative, we can find the solution to this problem. Now looking at this function, there's something that I notice. For the integrand, notice we have a 100 out in front and then we have this power function where we have x to the 99th. This is one reduced number from that number, so 100 minus 1 would be 99. So it looks like we went through some kind of power rule for the initial derivative. So what I'm going to do is try taking this front number out here, and I'm going to make it the exponent. So we're gonna have x to the 100th power. Now if I were to take this derivative, notice we would get 100 x to the 99th power. So this would give us the integrand. So this right here would be the function that we're looking for. This is the antiderivative, and don't forget to add the constant of integration c. So that right there is the solution to this problem. Let's move on.

Find the following indefinite integral. ∫ 100 x 99 d x \int100x^{99}dx ∫ 100 x 99 d x

100 99 x 100 \frac{100}{99}x^{100} 99 100 x 100

100 99 x 100 + C \frac{100}{99}x^{100}+C 99 100 x 100 + 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.
$\int100x^{99}dx$

$\frac{100}{99}x^{100}$

$x^{100}$

$\frac{100}{99}x^{100}+C$

$x^{100}+C$

Verified step by step guidance

Step 1: Recognize that the problem involves finding the indefinite integral of the function 100x^99. The general formula for integrating a power function x^n is ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1.

Step 2: Apply the formula to the given function. Here, n = 99, so we increase the exponent by 1 (n + 1 = 100) and divide by the new exponent (100). This gives us (100x^(100))/100 + C.

Step 3: Simplify the coefficient. The 100 in the numerator and denominator cancel out, leaving x^100 + C.

Step 4: Write the final expression for the indefinite integral, which is x^100 + C. Remember, C is the constant of integration that accounts for any constant term that could have been differentiated to give the original function.

Step 5: Verify the result by differentiating x^100 + C. The derivative of x^100 is 100x^99, and the derivative of C is 0. This confirms that the integral is correct.