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Notes
Questions
Find each indefinite integral
\(\textbf{1)}\) \(\displaystyle \int x+5 \,dx\)
\(\textbf{2)}\) \(\displaystyle \int x^4-2x^2+5x \,dx\)
\(\textbf{3)}\) \(\displaystyle \int x^{1.5}+5x \,dx\)
\(\textbf{4)}\) \(\displaystyle \int \displaystyle \frac{1}{x^5} \,dx\)
\(\textbf{5)}\) \(\displaystyle \int \sqrt[3]{x}+2 \,dx\)
\(\textbf{6)}\) \(\displaystyle \int \displaystyle \frac{x^3+5x^2-4}{\sqrt{x}} \,dx\)
\(\textbf{7)}\)\(\displaystyle \int x^3{\sqrt{x}} \,dx\)
\(\textbf{8)}\)\(\displaystyle \int dx\)
\(\textbf{9)}\) \(\displaystyle \int \frac{1}{x} \,dx\)
\(\textbf{10)}\) \(\displaystyle \int e^x \,dx\)
\(\textbf{11)}\) \(\displaystyle \int \sin x \,dx\)
\(\textbf{12)}\) \(\displaystyle \int \cos x \,dx\)
\(\textbf{13)}\) \(\displaystyle \int \sec^2 x \,dx\)
\(\textbf{14)}\) \(\displaystyle \int \csc^2 x \,dx\)
\(\textbf{15)}\) \(\displaystyle \int \sec x \tan x \,dx\)
\(\textbf{16)}\) \(\displaystyle \int \csc x \cot x \,dx\)
\(\textbf{17)}\) \(\displaystyle \int \frac{1}{1+x^2} \,dx\)
\(\textbf{18)}\) \(\displaystyle \int \frac{1}{\sqrt{1-x^2}} \,dx\)
See Related Pages\(\)
\(\bullet\text{ Indefinite Integral Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Trapezoidal Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{b-a}{2n}\left[f(a)+2f(x_1)+2f(x_2)+…+2fx_{n-1}+f(b)\right]…\)
\(\bullet\text{ Properties of Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}cf(x) \, dx=c\displaystyle \int_{a}^{b}f(x) \,dx…\)
\(\bullet\text{ Indefinite Integrals- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int x^n \, dx = \displaystyle \frac{x^{n+1}}{n+1}+C…\)
\(\bullet\text{ Indefinite Integrals- Trig Functions}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int \cos{x} \, dx=\sin{x}+C…\)
\(\bullet\text{ Definite Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{5}^{7} x^3 \, dx…\)
\(\bullet\text{ Integration by Substitution}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int (x^2+3)^3(2x) \,dx…\)
\(\bullet\text{ Area of Region Between Two Curves}\)
\(\,\,\,\,\,\,\,\,A=\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]\,dx…\)
\(\bullet\text{ Arc Length}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}\sqrt{1+\left[f'(x)\right]^2} \,dx…\)
\(\bullet\text{ Average Function Value}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{1}{b-a} \int_{a}^{b}f(x) \,dx\)
\(\bullet\text{ Volume by Cross Sections}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Disk Method}\)
\(\,\,\,\,\,\,\,\,V=\displaystyle \int_{a}^{b}\left[f(x)\right]^2\,dx…\)
\(\bullet\text{ Cylindrical Shells}\)
\(\,\,\,\,\,\,\,\,V=2 \pi \displaystyle \int_{a}^{b} y f(y) \, dy…\)
In Summary
Indefinite integrals, also known as antiderivatives, are a key concept in calculus. Eventually we will work our way up to finding definite integrals which will enable us to do many more things with functions, for example, finding the area under a curve.
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