Integration

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Q1. Find the integral of (2x + 3)dx. (2022)

Solution:

Integrating term by term: ∫2xdx = x^2 and ∫3dx = 3x. Thus, ∫(2x + 3)dx = x^2 + 3x + C.

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Q2. What is the integral of x^n where n ≠ -1? (2021)

Solution:

The integral of x^n is (1/(n+1))x^(n+1) + C, provided n ≠ -1.

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Q3. Evaluate the integral ∫ (3x^2 + 2x) dx. (2020)

Solution:

The integral is (3/3)x^3 + (2/2)x^2 + C = x^3 + x^2 + C.

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Q4. Find ∫ (6x^5) dx. (2022)

Solution:

The integral is (6/6)x^6 + C = x^6 + C.

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Q5. Evaluate ∫(5x^4)dx. (2020)

Solution:

The integral of 5x^4 is (5/5)x^5 + C = x^5 + C.

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Q6. Find the integral of sin(x). (2020)

Solution:

The integral of sin(x) is -cos(x) + C.

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Q7. Find the integral of cos(x)dx. (2023)

Solution:

The integral of cos(x) is sin(x) + C.

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Q8. What is the integral of x^2 with respect to x? (2021)

Solution:

The integral of x^n is (1/(n+1))x^(n+1) + C. Here, n=2, so the integral is (1/3)x^3 + C.

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Q9. Evaluate the integral ∫(3x^2 + 2)dx. (2022)

Solution:

Integrating term by term, ∫3x^2dx = x^3 and ∫2dx = 2x. Thus, ∫(3x^2 + 2)dx = x^3 + 2x + C.

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Q10. Evaluate ∫(2x^2 + 3x + 1)dx. (2021)

Solution:

Integrating term by term: ∫2x^2dx = (2/3)x^3, ∫3xdx = (3/2)x^2, and ∫1dx = x. Thus, ∫(2x^2 + 3x + 1)dx = (2/3)x^3 + (3/2)x^2 + x + C.