Cards (69)

  • What is the antiderivative of x^{3}</latex>?
    x44+\frac{x^{4}}{4} +C C
  • To integrate 5x25x^{2}, we can factor out the constant 5
  • The power rule of integration includes a constant of integration
  • What is the antiderivative of 3x23x^{2}?

    x3+x^{3} +C C
  • Steps to integrate sin(2x)dx\int \sin(2x) \, dx
    1️⃣ Let u=u =2x 2x
    2️⃣ Find du=du =2dx 2 \, dx
    3️⃣ Solve for dx=dx =du2 \frac{du}{2}
    4️⃣ Substitute into the integral
    5️⃣ Integrate to get 12cos(u)+- \frac{1}{2} \cos(u) +C C
    6️⃣ Replace uu with 2x2x
    7️⃣ Simplify to 12cos(2x)+- \frac{1}{2} \cos(2x) +C C
  • What is the integral of 2ex2e^{x}?

    2ex+2e^{x} +C C
  • The constant multiple rule states kf(x)dx=\int k \cdot f(x) \, dx =kf(x)dx k \int f(x) \, dx.

    True
  • Match the trigonometric function with its integral:
    sin(x)\sin(x) ↔️ cos(x)+- \cos(x) +C C
    cos(x)\cos(x) ↔️ sin(x)+\sin(x) +C C
    tan(x)\tan(x) ↔️ lnsec(x)+\ln|\sec(x)| +C C
    cot(x)\cot(x) ↔️ lnsin(x)+\ln|\sin(x)| +C C
  • Steps to integrate \int \sin(3x) \, dx</latex>
    1️⃣ Let u=u =3x 3x, then du=du =3dx 3 \, dx and dx=dx =du3 \frac{du}{3}
    2️⃣ Rewrite the integral: 13sin(u)du\frac{1}{3} \int \sin(u) \, du
    3️⃣ Integrate: 13cos(u)+- \frac{1}{3} \cos(u) +C C
    4️⃣ Substitute back: 13cos(3x)+- \frac{1}{3} \cos(3x) +C C
  • What is the integral of tan(x)\tan(x)?

    lnsec(x)+\ln|\sec(x)| +C C
  • Steps to integrate tan(x)\tan(x)
    1️⃣ Rewrite tan(x)=\tan(x) =sin(x)cos(x) \frac{\sin(x)}{\cos(x)}
    2️⃣ Let u=u =cos(x) \cos(x), so du=du =sin(x)dx - \sin(x) \, dx
    3️⃣ Rewrite the integral as 1udu- \int \frac{1}{u} \, du
    4️⃣ Integrate to get lnu+- \ln|u| +C C
    5️⃣ Substitute back to lnsec(x)+\ln|\sec(x)| +C C
  • The power rule of integration states that xndx=\int x^{n} \, dx =xn+1n+1+ \frac{x^{n + 1}}{n + 1} +C C, where n1n \neq - 1 and CC is the constant of integration
  • What is the constant multiple rule of integration?
    \int k \cdot f(x) \, dx = k \int f(x) \, dx</latex>
  • The integral of (x^{2} + 3x)</latex> is x33+\frac{x^{3}}{3} +3x22+ \frac{3x^{2}}{2} +C C.

    True
  • The constant multiple rule states that \int k \cdot f(x) \, dx = k \int f(x) \, dx</latex>.

    True
  • What is the antiderivative of cos(x)\cos(x)?

    sin(x)+\sin(x) +C C
  • What is the integral of cos(3x)\cos(3x)?

    13sin(3x)+\frac{1}{3} \sin(3x) +C C
  • What is the integral of x4x^{4}?

    x55+\frac{x^{5}}{5} +C C
  • What is the integral of (x2+(x^{2} +3x) 3x)?

    x33+\frac{x^{3}}{3} +3x22+ \frac{3x^{2}}{2} +C C
  • What is the integral of tan(x)\tan(x)?

    lnsec(x)+\ln|\sec(x)| +C C
  • The integral of cos(x)\cos(x) is \sin(x)
  • The integral of sin(u)\sin(u) is - \cos(u)
  • What is the constant of integration represented by in the integral of exe^{x}?

    C
  • Match the term with its original function:
    exdx\int e^{x} \, dx ↔️ ex+e^{x} +C C
    3exdx\int 3e^{x} \, dx ↔️ 3ex+3e^{x} +C C
  • When integrating ln(x)\ln(x), the expression xln(x)xx \ln(x) - x is part of the antiderivative.

    True
  • What is the Constant Multiple Rule in integration?
    kf(x)dx=\int k \cdot f(x) \, dx =kf(x)dx k \int f(x) \, dx
  • What does the Constant Multiple Rule state in integration?
    Constant times integral of function
  • Which rule is used to integrate x2x^{2} in the example?

    Power Rule
  • Write the general formula for the Constant Multiple Rule.
    \int k \cdot f(x) \, dx = k \int f(x) \, dx</latex>
  • What is the integral of exe^{x} without the Constant Multiple Rule?

    ex+e^{x} +C C
  • The integral of 3ex3e^{x} is 3ex+3e^{x} +C C
    True
  • Steps to integrate 3ex3e^{x} using the Constant Multiple Rule and Exponential Rule.

    1️⃣ Apply the Constant Multiple Rule: 3exdx=\int 3e^{x} \, dx =3exdx 3 \int e^{x} \, dx
    2️⃣ Integrate exe^{x} using the Exponential Rule: 3ex+3e^{x} +C C
  • The Sum and Difference Rule allows you to integrate terms separately.
    True
  • Match the function type with the appropriate integration rule:
    Power Function ↔️ Power Rule
    Constant Multiple Function ↔️ Constant Multiple Rule
    Sum/Difference of Functions ↔️ Sum/Difference Rule
    Trigonometric Function ↔️ Trigonometric Integral Rules
    Exponential Function ↔️ Exponential Integration Rule
  • cos(x)dx=\int \cos(x) \, dx =sin(x)+ \sin(x) +C C is the integral of a trigonometric function
  • The power rule of integration is valid when n=n =1 - 1.

    False
  • What does the sum and difference rule of integration state?
    (f(x)±g(x))dx=\int (f(x) \pm g(x)) \, dx =f(x)dx±g(x)dx \int f(x) \, dx \pm \int g(x) \, dx
  • What is the antiderivative of x4x^{4}?

    x55+\frac{x^{5}}{5} +C C
  • The integral of sin(x)\sin(x) is -\cos(x) + C</latex>, where CC is the constant of integration
  • When using u-substitution for sin(2x)dx\int \sin(2x) \, dx, du=du =2dx 2 \, dx.

    True