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Notes
Practice Problems
Find the integral
\(\textbf{1)}\)\(\displaystyle \int (x^2+3)^3(2x) \,dx\)
\(\textbf{2)}\)\(\displaystyle \int x^2(5x^3+3)^2 \,dx\)
\(\textbf{3)}\)\(\displaystyle \int \frac{\sin x}{\cos^4 x} \,dx\)
\(\textbf{4)}\)\(\displaystyle \int x \sin (x^2) \,dx\)
\(\textbf{5)}\)\(\displaystyle \int \frac{8x^4}{\sqrt{x^5-2}} \,dx\)
\(\textbf{6)}\)\(\displaystyle \int 5x^2 \sqrt{x^3+10} \,dx\)
\(\textbf{7)}\)\(\displaystyle \int \frac{\sec^2 (\frac{1}{x^6})}{x^7} \,dx\)
\(\textbf{8)}\)\(\displaystyle \int \sec^2 x \tan^2 x \,dx\)
\(\textbf{9)}\)\(\displaystyle \int e^x \sqrt{15+e^x} \,dx\)
\(\textbf{10)}\)\(\displaystyle \int e^{\cos 5 \theta} \sin 5\theta \,d\theta\)
\(\textbf{11)}\)\(\displaystyle \int \frac{\sin (\ln(4x))}{x} \,dx\)
\(\textbf{12)}\)\(\displaystyle \int x e^{x^2} \,dx\)
See Related Pages\(\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Trapezoidal Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{b-a}{2n}\left[f(a)+2f(x_1)+2f(x_2)+…+2fx_{n-1}+f(b)\right]…\)
\(\bullet\text{ Properties of Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}cf(x) \, dx=c\displaystyle \int_{a}^{b}f(x) \,dx…\)
\(\bullet\text{ Indefinite Integrals- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int x^n \, dx = \displaystyle \frac{x^{n+1}}{n+1}+C…\)
\(\bullet\text{ Indefinite Integrals- Trig Functions}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int \cos{x} \, dx=\sin{x}+C…\)
\(\bullet\text{ Definite Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{5}^{7} x^3 \, dx…\)
\(\bullet\text{ Integration by Substitution}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int (x^2+3)^3(2x) \,dx…\)
\(\bullet\text{ Area of Region Between Two Curves}\)
\(\,\,\,\,\,\,\,\,A=\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]\,dx…\)
\(\bullet\text{ Arc Length}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}\sqrt{1+\left[f'(x)\right]^2} \,dx…\)
\(\bullet\text{ Average Function Value}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{1}{b-a} \int_{a}^{b}f(x) \,dx\)
\(\bullet\text{ Volume by Cross Sections}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Disk Method}\)
\(\,\,\,\,\,\,\,\,V=\displaystyle \int_{a}^{b}\left[f(x)\right]^2\,dx…\)
\(\bullet\text{ Cylindrical Shells}\)
\(\,\,\,\,\,\,\,\,V=2 \pi \displaystyle \int_{a}^{b} y f(y) \, dy…\)
In Summary
Integration by Substitution (aka “u substitution”) is a method for simplifying certain tricky integrals (or antiderivatives.) It involves substituting a new variable for a certain part of the function being integrated. Usually you want to set it up so that the entire integral can be expressed in terms of your new variable “u” and the derivative of your new variable, “du.” They are commonly used with Trig functions, exponential functions, logarithmic functions, polynomial functions, and more.
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