Test Questions on the Indefinite Integral Symbol and Integrand

📚 Quick Study Guide

• 🔍 The indefinite integral symbol, denoted by $\int$, represents the antiderivative of a function.
• 💡 The integrand is the function under the integral sign, i.e., the function you're finding the antiderivative of. For example, in $\int f(x) dx$, $f(x)$ is the integrand.
• 📝 The $dx$ indicates the variable with respect to which the integration is performed.
• ➕ Remember to add the constant of integration, $C$, to every indefinite integral because the derivative of a constant is zero. So, $\int f(x) dx = F(x) + C$, where $F'(x) = f(x)$.
• 🍎 Basic Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.
• 📈 Integral of $e^x$: $\int e^x dx = e^x + C$.
• 📐 Integral of Trigonometric Functions:
  • $\int \sin(x) dx = -\cos(x) + C$
  • $\int \cos(x) dx = \sin(x) + C$

Practice Quiz

1. What is the indefinite integral symbol?
   1. $\sum$
   2. $\int$
   3. $\frac{d}{dx}$
   4. $\lim$
2. In the expression $\int (x^2 + 1) dx$, what is the integrand?
   1. $x$
   2. $dx$
   3. $x^2 + 1$
   4. $\int$
3. What does 'dx' signify in the expression $\int f(x) dx$?
   1. The derivative of x
   2. The differential of x, indicating the variable of integration
   3. The domain of f
   4. The definite integral
4. Why is the constant of integration 'C' added to indefinite integrals?
   1. To make the integral definite
   2. Because the derivative of a constant is always zero
   3. To satisfy initial conditions
   4. Because integrals are always approximate
5. What is the indefinite integral of $x^3$ with respect to $x$?
   1. $3x^2 + C$
   2. $\frac{x^4}{4} + C$
   3. $x^4 + C$
   4. $\frac{x^3}{3} + C$
6. What is the indefinite integral of $e^x$ with respect to $x$?
   1. $xe^x + C$
   2. $e^x + C$
   3. $\frac{e^{x+1}}{x+1} + C$
   4. $1 + C$
7. What is the indefinite integral of $\cos(x)$ with respect to $x$?
   1. $-\sin(x) + C$
   2. $\sin(x) + C$
   3. $-\cos(x) + C$
   4. $\tan(x) + C$