Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?
Problems & Solutions
Find the antiderivative of each function.
\(\textbf{1)}\) \(\displaystyle f(x)= x+5 \)
\(\textbf{2)}\) \(\displaystyle f(x)= x^4-2x^2+5x \)
\(\textbf{3)}\) \(\displaystyle f(x)= x^{1.5}+5x \)
\(\textbf{4)}\) \(\displaystyle f(x)= \displaystyle \frac{1}{x^5} \)
\(\textbf{5)}\) \(\displaystyle f(x)= \sqrt[3]{x}+2 \)
\(\textbf{6)}\) \(\displaystyle f(x)= \displaystyle \frac{x^3+5x^2-4}{\sqrt{x}} \)
\(\textbf{7)}\) \(\displaystyle f(x)= x^3{\sqrt{x}} \)
\(\textbf{8)}\) \(\displaystyle f(x)= 0 \)
See Related Pages\(\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Trapezoidal Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{b-a}{2n}\left[f(a)+2f(x_1)+2f(x_2)+…+2fx_{n-1}+f(b)\right]…\)
\(\bullet\text{ Properties of Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}cf(x) \, dx=c\displaystyle \int_{a}^{b}f(x) \,dx…\)
\(\bullet\text{ Indefinite Integrals- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int x^n \, dx = \displaystyle \frac{x^{n+1}}{n+1}+C…\)
\(\bullet\text{ Indefinite Integrals- Trig Functions}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int \cos{x} \, dx=\sin{x}+C…\)
\(\bullet\text{ Definite Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{5}^{7} x^3 \, dx…\)
\(\bullet\text{ Integration by Substitution}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int (x^2+3)^3(2x) \,dx…\)
\(\bullet\text{ Area of Region Between Two Curves}\)
\(\,\,\,\,\,\,\,\,A=\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]\,dx…\)
\(\bullet\text{ Arc Length}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}\sqrt{1+\left[f'(x)\right]^2} \,dx…\)
\(\bullet\text{ Average Function Value}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{1}{b-a} \int_{a}^{b}f(x) \,dx…\)
\(\bullet\text{ Volume by Cross Sections}\)
\(\,\,\,\,\,\,\,\,\text{Volume}=\displaystyle \int_{a}^{b}\left(\text{Area}\right) \, dx…\)
\(\bullet\text{ Disk Method}\)
\(\,\,\,\,\,\,\,\,V=\displaystyle \int_{a}^{b}\left[f(x)\right]^2\,dx…\)
\(\bullet\text{ Cylindrical Shells}\)
\(\,\,\,\,\,\,\,\,V=2 \pi \displaystyle \int_{a}^{b} y f(y) \, dy…\)
In Summary
An antiderivative, also known as an indefinite integral, is a mathematical concept used to find the original function given its derivative. It is the reverse process of taking a derivative.
Antiderivatives are related to integrals and are an important in mathematics because they allow us to solve problems involving the accumulation of quantities, such as distance. They are used in fields such as physics, engineering, and economics to model real-world phenomena.
Antiderivatives are typically introduced in a calculus course. They are a fundamental concept in calculus, and are used throughout the subject.
Common mistakes made when working with antiderivatives include forgetting to add the constant (+C).
Real world examples of antiderivatives
Distance traveled: The antiderivative of speed (rate of change of distance with respect to time) is distance traveled.
Work done: The antiderivative of force (rate of change of work with respect to distance) is work done.
Electric charge: The antiderivative of electric current (rate of change of electric charge with respect to time) is electric charge.
Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.
Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.
Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!
Thank you for visiting. How exciting!