Find f(x)f\left(x\right)f(x) by evaluating the following indefinite integral.f(x)=∫(8x7+10x−20)dxf\left(x\right)=\int\left(8x^7+10x-20\right)dx

f ( x ) = x 8 + 5 x 2 − 20 x + C f\left(x\right)=x^8+5x^2-20x+C

Find f ( x ) f\left(x\right) f ( x ) by evaluating the following indefinite integral. f ( x ) = ∫ ( 8 x 7 + 10 x − 20 ) d x f\left(x\right)=\int\left(8x^7+10x-20\right)dx

this problem, we're asked to find the function f of x by evaluating the following indefinite integral. Now to solve this problem, I'm going to use the rules for integration that I know. So what I can do first is I can take this integral, and I can integrate each term that I see where whether we're adding or subtracting the term. So we'll have the integral of 8 x to the 7th power, is gonna be plus the integral of 10 x with respect to x, and this is going to be minus the integral of 20 with respect to x. And I should also have the d x on this term here. So notice how we've just taken this integral, and we've basically just distributed it into each term that we have. So that's what we can do with these integrals, that's the rule. Now what I can also do is take any constants I see and pull them outside of the integral. So we have this 8 and this 10, and I could pull this 20 outside of the integral also, but since that constant is by itself, there's no x terms or anything, I'm just gonna leave it inside the integral. So doing this, we'll have 8 times the integral of x to the 7th with respect to x, then we're going to have plus 10 times the integral of x with respect to x, They're going to have minus the integral of 20 with respect to x. Now we can go ahead and integrate each one of these terms using the power rule. So we're going to have 8 times the integral of x to the 7th, and what I can do is use the power rule here. Add 1 to the exponent and divide by the exponent. That's gonna give me x to the 8th over 8. Then we're gonna have plus the integral of 10 times this integral right here, which we're going to use the same power rule that we have. So going to have x squared over 2, then we're going to have minus the integral of 20, which is going to be 20 x, then don't forget to add the plus c at the end. Now a couple things I can do to simplify this function that we just calculated, we can cancel the eights right here, and I can also take this 10 and divide it by 2, which will give me 5. So rewriting this function over here, we're going to have x to the 8th power. That's going to be plus 5 x squared. So we're just writing out this this function that we had down here. So we have plus 5 x squared. You're going to have minus 20x plus the constant of integration c, and this right here would be the solution to our problem. So this is how you can do this indefinite integral where we have multiple terms in this polynomial. Hope you found this video helpful. Let's get some more practice.

Find f ( x ) f\left(x\right) f ( x ) by evaluating the following indefinite integral. f ( x ) = ∫ ( 8 x 7 + 10 x − 20 ) d x f\left(x\right)=\int\left(8x^7+10x-20\right)dx

f ( x ) = x 8 + 5 x 2 − 20 x + C f\left(x\right)=x^8+5x^2-20x+C

f ( x ) = 8 x 8 + 10 x 2 − 20 x + C f\left(x\right)=8x^8+10x^2-20x+C

f ( x ) = x 8 + 5 x 2 + C f\left(x\right)=x^8+5x^2+C

f ( x ) = 8 x 8 + 10 x 2 + C f\left(x\right)=8x^8+10x^2+C

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

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Multiple Choice

Find $f\left(x\right)$ by evaluating the following indefinite integral.
$f (x) = \int (8 x^{7} + 10 x - 20) d x$

$f (x) = x^{8} + 5 x^{2} - 20 x + C$

$f (x) = 8 x^{8} + 10 x^{2} - 20 x + C$

$f (x) = x^{8} + 5 x^{2} + C$

$f (x) = 8 x^{8} + 10 x^{2} + C$

Verified step by step guidance

Identify the integral to be evaluated: $ \int (8x^7 + 10x - 20) \, dx $.

Apply the power rule for integration, which states that $ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $, to each term of the polynomial separately.

For the first term $ 8x^7 $, integrate to get $ \frac{8x^{8}}{8} = x^8 $.

For the second term $ 10x $, integrate to get $ \frac{10x^{2}}{2} = 5x^2 $.

For the constant term $-20$, integrate to get $-20x$. Combine all terms and add the constant of integration $ C $ to get the final expression: $ f(x) = x^8 + 5x^2 - 20x + C $.

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