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

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

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

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

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

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

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

Start by identifying the function to integrate: $ f(x) = \int (100x^2 - 35x - \frac{13}{2}) \, dx $.

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

Integrate each term separately: For $ 100x^2 $, use $ \int 100x^2 \, dx = \frac{100}{3}x^3 $. For $ -35x $, use $ \int -35x \, dx = -\frac{35}{2}x^2 $. For $ -\frac{13}{2} $, use $ \int -\frac{13}{2} \, dx = -\frac{13}{2}x $.

Combine the integrated terms: $ f(x) = \frac{100}{3}x^3 - \frac{35}{2}x^2 - \frac{13}{2}x + C $.

Remember to include the constant of integration $ C $ in your final expression, as indefinite integrals always include this constant.

5:04m

Watch next

Master Introduction to Indefinite Integrals with a bite sized video explanation from Patrick

Start learning

Indefinite Integrals practice set

Problem sets built by lead tutors

Expert video explanations

Go to Practice