Find h(x)h\left(x\right)h(x) by evaluating the following indefinite integral.h(x)=x100dxh\left(x\right)=x^{100}dxh(x)=x100dx

h ( x ) = x 101 101 + C h\left(x\right)=\frac{x^{101}}{101}+C h ( x ) = 101 x 101 + C

Find h ( x ) h\left(x\right) h ( x ) by evaluating the following indefinite integral. h ( x ) = x 100 d x h\left(x\right)=x^{100}dx h ( x ) = x 100 d x

here we're asked to find the function h by evaluating the following integral. Now in this problem, we have the integral of x to the 100th power with respect to x. Now to do this integral, recall that whenever we have these power functions that we see like this, what we can do is use the power rule, where we add 1 to the exponent and then divide the whole function, the the integrand here, by that new exponent. So doing this will give me for this integral, x to the 100 plus 1, which is a 101, and all of that's going to be divided by a 100+1, which is a 101. And then don't forget the constant of integration c that you have to add at the end anytime you do an indefinite integral. So this right here would be the solution for our function h of x, and that's how you can solve this problem. Let's move on.

Find h ( x ) h\left(x\right) h ( x ) by evaluating the following indefinite integral. h ( x ) = x 100 d x h\left(x\right)=x^{100}dx h ( x ) = x 100 d x

h ( x ) = x 101 101 + C h\left(x\right)=\frac{x^{101}}{101}+C h ( x ) = 101 x 101 + C

h ( x ) = x 101 100 + C h\left(x\right)=\frac{x^{101}}{100}+C h ( x ) = 100 x 101 + C

h ( x ) = x 101 101 h\left(x\right)=\frac{x^{101}}{101} h ( x ) = 101 x 101

h ( x ) = x 101 100 h\left(x\right)=\frac{x^{101}}{100} h ( x ) = 100 x 101

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

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Multiple Choice

Find $h\left(x\right)$ by evaluating the following indefinite integral.
$h\left(x\right)=x^{100}dx$

$h\left(x\right)=\frac{x^{101}}{101}+C$

$h\left(x\right)=\frac{x^{101}}{100}+C$

$h\left(x\right)=\frac{x^{101}}{101}$

$h\left(x\right)=\frac{x^{101}}{100}$

Verified step by step guidance

Identify the integral to be solved: $ \int x^{100} \, dx $. This is an indefinite integral, meaning we are looking for a function whose derivative is $ x^{100} $.

Recall the power rule for integration, which states that $ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $, where $ C $ is the constant of integration.

Apply the power rule to the integral $ \int x^{100} \, dx $. Here, $ n = 100 $, so we increase the exponent by 1 to get $ x^{101} $.

Divide by the new exponent, $ 101 $, to complete the integration process: $ \frac{x^{101}}{101} $.

Don't forget to add the constant of integration $ C $ to the result, as this is an indefinite integral: $ h(x) = \frac{x^{101}}{101} + C $.

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