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

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

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

Table of contents

Skip topic navigation

0. Functions7h 54m

Introduction to Functions
16m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
36m

Exponential & Logarithmic Equations
35m

Introduction to Trigonometric Functions
38m

Graphs of Trigonometric Functions
44m

Trigonometric Identities
47m

Inverse Trigonometric Functions
48m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation3h 18m

Basic Rules of Differentiation
39m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

Derivatives of Trig Functions
40m

The Chain Rule
49m

4. Applications of Derivatives2h 38m

Motion Analysis
36m

Implicit Differentiation
27m

Related Rates
59m

Linearization
13m

Differentials
21m

5. Graphical Applications of Derivatives6h 2m

Intro to Extrema
23m

Finding Global Extrema
50m

The First Derivative Test
1h 6m

Concavity
1h 1m

The Second Derivative Test
31m

Curve Sketching
44m

Applied Optimization
1h 25m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m

Derivatives of Exponential & Logarithmic Functions
1h 52m

Logarithmic Differentiation
20m

Derivatives of Inverse Trigonometric Functions
24m

7. Antiderivatives & Indefinite Integrals1h 26m

Antiderivatives
17m

Indefinite Integrals
25m

Integrals of Trig Functions
15m

Initial Value Problems
27m

8. Definite Integrals4h 44m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 21m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
37m

Average Value of a Function
21m

Substitution
1h 19m

9. Graphical Applications of Integrals2h 27m

Area Between Curves
1h 18m

Introduction to Volume & Disk Method
1h 8m

10. Physics Applications of Integrals 3h 16m

Kinematics
1h 13m

Work
2h 3m

11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m

Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

Integrals Involving Inverse Trigonometric Functions
55m

12. Techniques of Integration7h 41m

Integration by Parts
2h 32m

Partial Fractions
4h 29m

Improper Integrals
39m

13. Intro to Differential Equations2h 55m

Basics of Differential Equations
48m

Slope Fields
19m

Euler's Method
22m

Separable Differential Equations
1h 25m

14. Sequences & Series5h 36m

Sequences
1h 22m

Review of Factorials
11m

Series
44m

Convergence Tests
3h 17m

15. Power Series2h 19m

Introduction to Power Series
1h 9m

Taylor Series & Taylor Polynomials
1h 10m

16. Parametric Equations & Polar Coordinates7h 58m

Parametric Equations
1h 6m

Calculus with Parametric Curves
1h 13m

Polar Coordinates
2h 5m

Calculus in Polar Coordinates
55m

Conic Sections
2h 36m

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Previous problemNext problem

Struggling with Calculus?

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

Multiple Choice

Find the following indefinite integral.
$\int 100 x^{99} d x$

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

$x to the 100th power$

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

$x^{100} + C$

Previous problemNext problem

Verified step by step guidance

Identify the integral to be solved: ∫100x^{99}dx.

Apply the power rule for integration, which states that ∫x^n dx = (1/(n+1))x^{n+1} + C, where C is the constant of integration.

In this case, the integrand is 100x^{99}. The coefficient 100 can be factored out of the integral, simplifying the expression to 100∫x^{99}dx.

Use the power rule: ∫x^{99}dx = (1/(99+1))x^{99+1} = (1/100)x^{100}.

Multiply the result by the factored coefficient: 100 * (1/100)x^{100} = x^{100}. Add the constant of integration C to get the final expression: x^{100} + C.