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Quizzes > High School Quizzes > Mathematics

Integral Worksheet Practice Quiz

Sharpen calculus skills with guided quiz challenges

Difficulty: Moderate
Grade: Grade 12
Study OutcomesCheat Sheet
Paper art representing a trivia quiz challenging high school calculus students with integration problems

What is the indefinite integral of x^3?
3x^2 + C
x^2/2 + C
x^4/4 + C
x^3/3 + C
Using the power rule, ∫x^3 dx equals x^(3+1)/(3+1) = x^4/4 + C. The other options represent common mistakes when applying the power rule.
What is the antiderivative of 1 with respect to x?
1 + C
0
ln|x| + C
x + C
The integral of a constant 1 is x, plus the constant of integration. The other choices stem from common misunderstandings of basic integration.
Find the integral of 5 with respect to x.
x^5 + C
5 + C
5x + C
5x^5 + C
Integrating a constant involves multiplying it by x, thus ∫5 dx = 5x + C. The other options involve errors in applying the power rule.
Compute the integral ∫4x dx.
2x^2 + C
4x^2 + C
2x + C
x^2 + C
By applying the power rule, ∫4x dx equals 4*(x^2/2) = 2x^2 + C. The other choices do not correctly apply the integration process.
What is the antiderivative of cos(x)?
-sin(x) + C
-cos(x) + C
cos(x) + C
sin(x) + C
Since the derivative of sin(x) is cos(x), the antiderivative of cos(x) is sin(x) + C. The other options represent common sign and function errors.
Evaluate the definite integral from 0 to 2 of 3x^2 dx.
12
6
8
4
Integrating 3x^2 yields x^3, and evaluating from 0 to 2 gives 2^3 - 0 = 8. The other options arise from common computational errors.
What is the antiderivative of e^(x)?
ln|e^(x)| + C
e^(x) + C
e^(x) - 1 + C
xe^(x) + C
The function e^(x) is unique as its derivative and antiderivative are the same, giving e^(x) + C. The other options incorrectly incorporate additional factors.
Calculate ∫ (2x + 1) dx.
2x^2 + x + C
x^2/2 + x + C
2x + 1 + C
x^2 + x + C
Integrate term-by-term: ∫2x dx yields x^2 and ∫1 dx yields x, so the result is x^2 + x + C. The other answers represent misapplications of the integration rules.
Determine the integral ∫ x√(x^2 + 1) dx using an appropriate substitution.
1/2 (x^2 + 1)^(3/2) + C
2/3 (x^2 + 1)^(3/2) + C
(x^2 + 1)^(3/2) + C
1/3 (x^2 + 1)^(3/2) + C
Using the substitution u = x^2 + 1, the integral simplifies to 1/2∫u^(1/2) du, which evaluates to 1/3 (x^2 + 1)^(3/2) + C. The other options have incorrect constant factors.
What is the integral of sin(x) dx?
-sin(x) + C
cos(x) + C
sin(x) + C
-cos(x) + C
The integral of sin(x) is -cos(x) + C because the derivative of cos(x) is -sin(x). The other options reflect sign errors important in trigonometric integration.
Find the integral of 1/x dx.
x + C
ln|x| + C
e^x + C
1/(x^2) + C
The integral of 1/x is ln|x| + C, a fundamental result in calculus. The other choices are common misinterpretations of the integration process.
Evaluate ∫ 3e^(2x) dx.
e^(2x) + C
(3/2)e^(x) + C
3e^(2x) + C
(3/2)e^(2x) + C
Using the substitution u = 2x, we must adjust for the derivative, which produces a factor of 1/2. Hence, the correct answer is (3/2)e^(2x) + C, ensuring proper handling of the chain rule.
What is the antiderivative of sec^2(x)?
tan(x) + C
-sec(x) + C
-tan(x) + C
sec(x) + C
Since the derivative of tan(x) is sec^2(x), the antiderivative must be tan(x) + C. The other options do not differentiate to sec^2(x).
Find ∫ (x/(x^2 + 1)) dx.
ln|x^2| + C
½ ln|x^2 + 1| + C
1/(x^2 + 1) + C
ln|x| + C
Using the substitution u = x^2 + 1, with du = 2x dx, the integral becomes ½∫(1/u) du, resulting in ½ ln|x^2 + 1| + C. The other options misapply the logarithmic integration.
Calculate ∫ √x dx.
(2/3)x^(3/2) + C
(1/2)x^(3/2) + C
(3/2)x^(1/2) + C
x^(3/2) + C
Express √x as x^(1/2) and apply the power rule to get x^(3/2)/(3/2), which simplifies to (2/3)x^(3/2) + C. Other options fail to correctly invert the fractional exponent.
Evaluate the definite integral ∫ from 0 to π/4 of sec^2(x) dx.
π/4
1
√2
0
The antiderivative of sec^2(x) is tan(x). Evaluating tan(x) from 0 to π/4 gives tan(π/4) - tan(0) = 1 - 0 = 1. The other answers are results of misinterpretation of the limits.
Solve the integral ∫ ln(x) dx using integration by parts.
x ln(x) - x + C
ln(x) - x + C
x ln(x) - ln(x) + C
x ln(x) + x + C
Using integration by parts with u = ln(x) and dv = dx, the integral evaluates to x ln(x) - x + C. Incorrect options reflect sign errors or misuse of the parts formula.
Determine the antiderivative of x e^(x) using integration by parts.
e^(x)(x + 1) + C
x e^(x) + e^(x) + C
x e^(x) - e^(x) + C
e^(x)(x - 2) + C
Applying integration by parts with u = x and dv = e^(x) dx leads to x e^(x) - e^(x) + C. Other options arise from incorrect application of the method.
Find the indefinite integral ∫ tan(x) dx.
ln|cos(x)| + C
- ln|cos(x)| + C
sec(x) + C
-sec(x) + C
The integral of tan(x) is found by rewriting tan(x) as sin(x)/cos(x) and using substitution, resulting in -ln|cos(x)| + C. The alternatives come from common sign errors.
Compute the antiderivative ∫ [1/(x ln(x))] dx.
ln|x| + C
1/(x ln(x)) + C
ln|ln(x)| + C
1/ln(x) + C
Using the substitution u = ln(x), where du = 1/x dx, transforms the integral into ∫1/u du, which evaluates to ln|u| + C or ln|ln(x)| + C. The other answers do not correctly apply the substitution method.
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Study Outcomes

  1. Analyze dynamic integration problems using various techniques.
  2. Apply substitution and integration by parts to solve complex integrals.
  3. Evaluate definite and indefinite integrals to determine areas under curves.
  4. Interpret the results of integration problems to validate solutions.
  5. Synthesize multiple integration methods to approach composite functions efficiently.

Integral Worksheet Cheat Sheet

  1. Fundamental Theorem of Calculus - This theorem is the secret handshake between differentiation and integration, showing they undo each other. It's your ticket to evaluating definite integrals and understanding how quantities pile up over intervals. Wikipedia: Fundamental Theorem
  2. Basic Integration Formulas - Memorise key formulas like ∫x❿ dx = (1/(n+1))x❿❺¹ + C and ∫eˣ dx = eˣ + C to unlock a variety of problems. These building blocks make challenging integrals feel like a breeze once you get the hang of them. Practice them until they're second nature. Math10 Basic Integrals
  3. Integration by Substitution - Think of u-substitution as swapping puzzle pieces to simplify integrals. When you let u equal part of the integrand, the integral often collapses into a basic form. Try examples like ∫x cos(x²) dx to see the magic happen! GeeksforGeeks Substitution Practice
  4. Integration by Parts - This formula, ∫u dv = uv - ∫v du, is your go-to for products of functions. It transforms tough products into simpler terms, as shown by ∫x eˣ dx = x eˣ - eˣ + C. Keep swapping u and dv to become a pro. NRICH Integration & Differentiation Problems
  5. Trigonometric Integrals - Master integrals like ∫sin(x) dx = -cos(x) + C and ∫cos(x) dx = sin(x) + C for basic trig functions. For more complex combos, use identities such as sin²(x)+cos²(x)=1 to simplify first. With a bit of practice, these will feel like a breeze! Math10 Trigonometric Integrals
  6. Partial Fraction Decomposition - Break rational functions into simpler fractions using partial fractions, expressing them as sums like A/(x+1)+B/(x-1). This trick turns intimidating rational integrals into a series of easy steps. It's a classic method that never fails. NRICH Partial Fractions Practice
  7. Exponential & Logarithmic Integrals - Get comfy with integrals like ∫eˣ dx = eˣ + C and ∫ln(x) dx = x ln(x)-x + C for everything from growth models to entropy. These show up everywhere in calculus and beyond. Mastering them gives you a superpower in problem-solving! Math10 Exp & Log Integrals
  8. Inverse Trigonometric Integrals - Learn the antiderivatives: ∫dx/√(1-x²)=arcsin(x)+C and ∫dx/(1+x²)=arctan(x)+C. These are essential when square roots or quadratics lurk in denominators. They'll help you crack tricky integrals like a champ. Math10 Inverse Trig Integrals
  9. Strategy & Practice Variety - Solve a mix of definite and indefinite integrals to sharpen your problem-solving instincts. The more types you tackle, the faster you'll spot which technique fits. Build a varied workout routine for your brain to ace any integral challenge. BYJU'S Integration Questions
  10. Online Resources & Challenges - Level up with interactive problems and tutorials on sites packed with hints, solutions, and challenge levels. Regularly testing yourself online turns daunting topics into fun games. Dive in daily to solidify your skills and boost your confidence. OnlineMath4All Practice Questions
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