Find f(x)f\left(x\right)f(x) by evaluating the following indefinite integral.f(x)=∫(100x2−35x−132)dxf\left(x\right)=\int\left(100x^2-35x-\frac{13}{2}\right)dxf(x)=∫(100x2−35x−213)dx

f(x)=1003x3−352x2−132x+Cf\left(x\right)=\frac{100}{3}$$x^3$$-\frac{35}{2}$$x^2$$-\frac{13}{2}x+Cf(x)=3100x3−235x2−213x+C

Find f ( x ) f\left(x\right) f ( x ) by evaluating the following indefinite integral. f ( x ) = ∫ ( 100 x 2 − 35 x − 13 2 ) d x f\left(x\right)=\int\left(100x^2-35x-\frac{13}{2}\right)dx f ( x ) = ∫ ( 100 x 2 − 35 x − 2 13 ) d x

this problem, we're asked to evaluate the following indefinite integral and solve for this function f of x. Now notice we have the integral of this polynomial here. And when solving these types of problems, what we can do is apply the rules of integration that we've learned about. Now one of the rules I know for integrals is that I can take each term, and I can in essence distribute this integral to each term that we have whenever we're adding or subtracting. So let's go ahead and try this. So I'm going to integrate 100 x squared with respect to x, then we would have minus the integral of 35 x, Then we're going to have minus the integral of 13 over 2 with respect to x. So this is what happens when I integrate each one of these terms. Now another thing that I'm going to do, another rule that I can use is taking each constant that I see and pulling it outside of the integral. So we're going to have 100 times the integral of x squared, and we're going to have minus 35 times the integral of x. And I could technically take this 13 over 2 and pull it outside of the integral as well since that's also a constant. But because we have a constant by itself and it's not attached to any x variable, I'm just gonna go ahead and leave that inside the integral here. So we're gonna have 13 over 2 with respect to x. Usually, I like to take constants that are by themselves inside the integral and just leave them alone. So this is what we're gonna have. Now what I can do is go through here and integrate each one of these terms. Now recall that we learned the power rule for integrals. We can add 1 to the exponent and divide by that exponent. So we're going to have a 100, and it's going to be times, and then this is gonna be x cubed over 3. That's gonna be this first term here. We're going to have minus, and then we have 35, 35, and then I can use the power rule on this term as well. So we'll have to the 1 plus 1 over 1 plus 1, which is gonna give us x squared over 2. And then this is going to be minus the integral of 13 over 2 with respect to x, which is just 13 over 2x, and then don't forget the plus c, constant of integration at the end. So this right here would be the solution, and that is how we can solve this integral. Hope you found this video helpful, and let's move on and get some more practice.

Find f(x)f\left(x\right)f(x) by evaluating the following indefinite integral.f(x)=∫(100x2−35x−132)dxf\left(x\right)=\int\$$left(100x^2-35x$$-\frac{13}{2}\right)dxf(x)=∫(100x2−35x−213)dx

f(x)=1003x3−352x2−132x+Cf\left(x\right)=\frac{100}{3}$$x^3$$-\frac{35}{2}$$x^2$$-\frac{13}{2}x+Cf(x)=3100x3−235x2−213x+C

f(x)=50x3−35x2−132x+Cf\left(x\right)=$$50x^3$-35x^2$$-\frac{13}{2}x+Cf(x)=50x3−35x2−213x+C

f(x)=50x3−35x2+Cf\left(x\right)=$$50x^3$-35x^2+Cf(x)=50x3$$−35x2+C

f(x)=100x3−35x2−132x+Cf\left(x\right)=$$100x^3$-35x^2$$-\frac{13}{2}x+Cf(x)=100x3−35x2−213x+C

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Business Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

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Multiple Choice

Find $f$\left$(x$\right$)$ by evaluating the following indefinite integral.
$f$\left$(x$\right$)=$\int\[\left$(100x^2-35x-$\frac{13}{2}\]\right$)dx$

$f$\left$(x$\right$)=$\frac{100}{3}$x^3-$\frac{35}{2}$x^2-$\frac{13}{2}$x+C$

$f$\left$(x$\right$)=50x^3-35x^2-$\frac{13}{2}$x+C$

$f$\left$(x$\right$)=50x^3-35x^2+C$

$f$\left$(x$\right$)=100x^3-35x^2-$\frac{13}{2}$x+C$

Verified step by step guidance

Step 1: Recognize that the problem involves finding the indefinite integral of the given function. The integral is ∫(100x^2 - 35x - 13/2) dx. The goal is to integrate each term of the polynomial separately.

Step 2: Apply the power rule of integration to the first term, 100x^2. The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1. For 100x^2, n = 2, so the integral becomes (100x^(2+1))/(2+1) = (100x^3)/3.

Step 3: Apply the power rule of integration to the second term, -35x. Here, n = 1, so the integral becomes (-35x^(1+1))/(1+1) = (-35x^2)/2.

Step 4: Integrate the constant term, -13/2. The integral of a constant c is simply c * x. Therefore, the integral of -13/2 is (-13/2)x.

Step 5: Combine all the results from the previous steps and add the constant of integration, C. The final expression for f(x) is f(x) = (100x^3)/3 - (35x^2)/2 - (13/2)x + C.

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Struggling with Business Calculus?

Find f(x)f\(\left\)(x\(\right\))f(x) by evaluating the following indefinite integral.f(x)=∫(100x2−35x−132)dxf\(\left\)(x\(\right\))=\(\int\[\left\)(100x^2-35x-\(\frac{13}{2}\]\right\))dxf(x)=∫(100x2−35x−213)dx

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Find $f\(\left\)(x\(\right\))$ by evaluating the following indefinite integral.
$f\(\left\)(x\(\right\))=\(\int\[\left\)(100x^2-35x-\(\frac{13}{2}\]\right\))dx$