Integral Calculus Mock Test 2025: Practice Questions & Answers

Integration by substitution involves replacing a variable with a new variable to simplify the integral. Follow these steps:
1. Choose a substitution u = g(x) so that the integral becomes simpler.
2. Replace dx with du (using du = g'(x)dx).
3. Rewrite the integral in terms of u and du.
4. Integrate with respect to u.
5. Substitute back x into the final answer.
This technique is especially useful for integrals involving composite functions or chain rule derivatives.

The Integral Test can be applied if:
• The function f(x) is positive (i.e., f(x) > 0)
• The function is continuous for all x ≥ N
• The function is decreasing (i.e., f(x+1) ≤ f(x)) for all x ≥ N
Where N is some positive integer. These conditions ensure that the test provides an accurate conclusion for series convergence or divergence.

To test the convergence of the series ∑1/n², let f(x) = 1/x².
We check that f(x) is positive, continuous, and decreasing for x ≥ 1.
Perform the integral:
∫₁^∞ 1/x² dx = [ -1/x ]₁^∞ = 0 + 1 = 1
Since the integral converges, by the Integral Test, the series ∑1/n² also converges.

Indefinite integrals represent a general antiderivative and include a constant of integration (C).
Definite integrals compute the net area under a curve between two bounds (a and b), resulting in a specific numerical value and do not include the constant C. Both are fundamental concepts in the CBSE syllabus for calculus.

No, the Integral Test can only be applied to series whose nth-term function f(n) is positive, continuous, and decreasing for all n ≥ N. If the function does not meet these conditions, the test is not applicable and may give incorrect conclusions.

A series ∑aₙ is said to be convergent according to the Integral Test if the improper integral ∫ f(x) dx from N to ∞ converges, where aₙ = f(n). This means adding up infinitely many terms results in a finite sum if the corresponding area under the curve is also finite.

An improper integral is an integral with one or both limits infinite, or with an integrand that becomes infinite within the interval. In the Integral Test, we often evaluate ∫_N^∞ f(x) dx. If this value exists (converges), it helps determine series convergence.

Common mistakes include:
• Not checking whether the function is positive, continuous, and decreasing
• Incorrectly computing the improper integral
• Mixing up convergence of the series and the integral
• Forgetting to verify that the function matches the series term-for-term for large n
Careful validation prevents common exam errors.

To verify that f(x) is decreasing:
• Compute its derivative: If f'(x) ≤ 0 for all x ≥ N, the function is decreasing.
• You can also check values: If f(x+1) ≤ f(x) for all x ≥ N, it is decreasing.
This step is essential before using the Integral Test.

Yes, even if the individual terms aₙ approach zero, the Integral Test may still reveal that the series diverges. According to the test, it is the behavior of the improper integral (i.e., if it diverges), not just the term limit, that determines series divergence.

No, the Integral Test is not valid for series with negative, zero, or non-monotonic terms. The function must be positive, continuous, and decreasing for the test's assumptions to hold true and for the results to be accurate.