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Integrals — Class 12 solved problems

41 problems from this chapter, each solved step by step.

  1. Evaluate integral from 0 to pi of (x*sin(x)) / (1 + cos²(x)) dx and show that it equals pi² / 4.

    (π²)/(4)

  2. Evaluate the integral from 0 to pi/2 of sin^5(x) / (sin^5(x) + cos^5(x)) dx.

    The value of the integral is (π)/(4).

  3. Evaluate the integral of 1 / (x * (x^4 + 1)) dx.

    (1)/(4) ln|(x^4)/(x^4 + 1)| + C

  4. Evaluate the integral from 0 to pi/2 of (sin(x))^5 / ((sin(x))^5 + (cos(x))^5) dx.

    (π)/(4)

  5. Evaluate the integral of e^(2x) * cos(3x) dx.

    (1)/(13)e²x(2cos(3x) + 3sin(3x)) + C

  6. Evaluate the definite integral: integral from 0 to pi/2 of sin^4(x)*cos³(x) dx.

    (2)/(35)

  7. Evaluate the definite integral of (x² + 1) / (x^4 + 1) dx from 0 to infinity.

    (π)/(√2)

  8. 1. Definite Integral Property & Symmetry π ─── 2 √(sin x) Evaluate: ∫ ─────────────────── dx 0 √(sin x) + √(cos x)

    I = (π)/(4)

  9. Evaluate the definite integral ∫_0^((π)/(2)) (√(sin x))/(√(sin x) + √(cos x)) dx.

    The value of the definite integral is (π)/(4).

  10. If ∫_0^4 (dx)/(2x + 1) = log k, then the value of k is:

    3

  11. Assertion (A): ∫ (1)/(√(9 - x²)) dx = sin^(-1) (x)/(3). Reason (R): ∫ (1)/(√(a² - x²)) dx = sin^(-1) (x)/(a) + C.

    Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

  12. Find the flux of F⃗ = (x³, y³, z³) across the surface of the sphere x² + y² + z² = a².

    The flux of F across the surface of the sphere is (12π a^5)/(5).

  13. The integral 80 ∫_0^((π)/(2)) ((sin θ + cos θ)/(9 + 16 sin 2 θ)) dθ is equal to:

    8 ln 3

  14. Evaluate the double integral ∬∫∫_D e^(x²+y²) dx dy where D is the disk x² + y² ≤ 4.

    π (e^4 - 1)

  15. Find the value of the definite integral: ∫_0^(π) (x sin(x))/(1 + cos²(x)) dx

    (π²)/(4)

  16. Evaluate the integral of (x² + 1) / (x^4 + 1) dx from 0 to infinity.

    (π)/(√2)

  17. Let ∫ x³ sin x dx = g(x) + C, where C is the constant of integration. If 8(g((π)/(2)) + g'((π)/(2))) = α π³ + β π² +, α, β, Z, then α + β - equals:

    55

  18. The integral ∫_-1^((3)/(2)) | π² x sin(π x) | dx is equal to:

    3π + 1

  19. Evaluate ∫₀^(π/2) (sin x)^4 (cos x)³ dx using reduction formulas.

    (2)/(35)

  20. The integral ∫_0^(π)(8x)/(4cos²x + sin²x) dx is equal to:

    2π²

  21. If 24 ∫_0^((π)/(4)) (sin |4x - (π)/(12)| + |2 sin x|) dx = 2π + α, where [·] denotes the greatest integer function, then α is equal to:

    60 - 24√2 - 2π

  22. The value of ∫_-1^(1) ((1 + √(|x|) - x)e^x + (√(|x|) - x)e^(-x))/(e^x + e^(-x)) dx is equal to:

    (7)/(3)

  23. (b) Find: ∫ (x+3)/(x²+4x+5) dx (integral, quadratic, antiderivative, calculus, integration)

    (1)/(2) ln|x²+4x+5| + tan^(-1)(x+2) + C

  24. If ∫ e^x ( (x sin^(-1) x)/(√(1-x²)) + (x ln^(-1) x)/((1-x²)³/2) + (x)/(1-x²)) dx = g(x) + C, where C is the constant of integration, then g((1)/(2)) equals:

    (π e^(1/2))/(6√3)

  25. Evaluate ∬∫∫_E (x² + y²) dV where E is the region bounded by z = 1 - x² - y² and the xy-plane.

    The value of the integral is (π)/(6).

  26. VARIATION: Evaluate ∫₀^(π/2) (sin x)^4 (cos x)³ dx using reduction formulas. Find the numerical value if exact solution is not possible.

    (2)/(35)

  27. Let f(x) be a positive function and I_1 = ∫_-(1)/(2)^(1) 2x f(2x(1 - 2x)) dx and I_2 = ∫_-1^((1)/(2)) f(x(1 - x)) dx. Then the value of (I_2)/(I_1) is equal to:

    The value of (I_2)/(I_1) is equal to 2.

  28. Let f(x) = 7 tan^8 x + 7 tan^8 x - 3 tan^4 x - 3 tan² x, I_1 = ∫_0^(π/4) f(x) dx and I_2 = ∫_0^(π/4) x f(x) dx. Then 7 I_1 + 12 I_2 is equal to:

    12 I_2

  29. If I = ∫_0^((π)/(2)) (sin (1)/(2) x)/(sin² x + cos² x) dx, then ∫_0²I (x sin x cos x)/(sin² x + cos^4 x) dx equals:

    The problem statement is mathematically underdetermined due to an inconsistency in the upper limit of the second integral. If 2I is assumed to be π, the integra

  30. Let f be a continuous function on [a,b] such that ∫ₐᵇ xⁿf(x)dx = 0 for all n ≥ 0. Prove that f(x) = 0 for all x ∈ [a,b].

    Since ∫_a^(b) f(x)² dx = 0 and f(x) is continuous, it must be that f(x) = 0 for all x [a,b].

  31. Evaluate ∮_C (z² + 1)/(z² - 1) dz where C is the circle |z| = 3 using residue theorem.

    0

  32. Let f: (0, ∞) → R be a twice differentiable function. If for some a ≠ 0, ∫_0^1 f( x) d = a f(x), f(1) = 1 and f(16) = (1)/(8), then 16 - f'((1)/(16)) is equal t

    112

  33. If ∫ (2x² + 5x + 9)/(√(x² + x + 1)) dx = x √(x² + x + 1) + α √(x² + x + 1) + β log_e | x + (1)/(2) + √(x² + x + 1) | + C, where C is the constant of integration

    16

  34. Let f(x)+2f ((1)/(x))=x²+5 and 2g(x)-3g ((1)/(x))=x, x>0. If α=∫_1² f(x) dx and β=∫_1² g(x) dx, then the value of 9α+β is:

    19 + (3)/(5)(1 + ln 2)

  35. Let for some function y = f(x), ∫_0^x t f(t) dt = x² f(x), x > 0 and f(2) = 3. Then f(6) is equal to

    1

  36. Finally, we evaluate the definite integral of the line equation from x = 0 to x = 7. This calculates the area under the line segment PQ, which is the area of th

    The area of POQ is 21 square units.

  37. Evaluate the integral ∫₀^∞ ln(x)/(x² + 1) dx using complex analysis techniques.

    0

  38. Let (a, b) be the point of intersection of the curve x² = 2y and the straight line y - 2x - 6 = 0 in the second quadrant. Then the integral I = ∫_a^b (9x²)/(1 +

    24

  39. 4∫_0^1 ((1)/(√(3 + x²) + √(1 + x²))) dx - 3 log_e(√3) is equal to:

    2 - √2 - log_e(1 + √2)

  40. If ∫_-(π)/(2)^((π)/(2)) (25 x² cos² x)/(2(x + e)²) dx = π (α π² + β), α, β Z, then (α + β)² equals

    The problem statement appears to be incorrect or have a typo, as the integral does not simplify to the given form using standard techniques.

  41. Let ⌊·⌋ denote the greatest integer function. If ∫_0^e³ (1)/(e^(x-1)) dx = α - ln 2, then α³ is equal to:

    8

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