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Solutions

Overview

Step-by-step NCERT solutions for Integrals (Chapter 7, CBSE Class 12 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Integrals textbook chapter.

Solved

What these solutions cover

All 261 questions in Integrals are solved in the PDF. Here's what's inside, exercise by exercise:

Integration by Inspection

  1. Find the integral: ∫ sin 2x dx
  2. Find the integral: ∫ cos 3x dx
  3. Find the integral: ∫ e²ˣ dx
  4. Find the integral: ∫ (ax + b)² dx
  5. Find the integral: ∫ sin 2x − 4e³ˣ dx
  6. Find the integral: ∫ (4e³ˣ + 1) dx
  7. Find the integral: ∫ x² (1 − 1/x²) dx
  8. Find the integral: ∫ (ax² + bx + c) dx
  9. Find the integral: ∫ (2x² + eˣ) dx
  10. Find the integral: ∫ (√x − 1/√x)² dx
  11. Find the integral: ∫ (x³ + 5x² − 4) / x² dx
  12. Find the integral: ∫ (x³ + 3x + 4) / √x dx
  13. Find the integral: ∫ (x³ − x² + x − 1) / (x − 1) dx
  14. Find the integral: ∫ (1 − x) √x dx
  15. Find the integral: ∫ √x (3x² + 2x + 3) dx
  16. Find the integral: ∫ (2x − 3 cos x + eˣ) dx
  17. Find the integral: ∫ (2x² − 3 sin x + 5√x) dx
  18. Find the integral: ∫ sec x (sec x + tan x) dx
  19. Find the integral: ∫ (sec²x) / cosec²x dx
  20. Find the integral: ∫ (2 − 3 sin x) / cos²x dx
  21. Choose the correct answer: The anti derivative of (√x + 1/√x) equals (A) (1/3)x^(1/3) + 2x^(1/2) + C (B) (2/3)x^(2/3) + (1/2)x^(1/2) + C (C) (2/3)x^(3/2) + 2x^(1/2) + C (D) (3/2)x^(3/2) + (1/2)x^(1/2) + C
  22. Choose the correct answer: If d/dx f(x) = 4x³ − 3/x⁴ such that f(2) = 0, then f(x) is (A) x⁴ + 1/x³ − 129/8 (B) x³ + 1/x⁴ + 129/8 (C) x⁴ + 1/x³ + 129/8 (D) x³ + 1/x⁴ − 129/8

Integration by Substitution

  1. Find the integral: ∫ 2x / (1 + x²) dx
  2. Find the integral: ∫ (log x)² / x dx
  3. Find the integral: ∫ 1 / (x + x log x) dx
  4. Find the integral: ∫ sin x · sin(cos x) dx
  5. Find the integral: ∫ sin(ax + b) cos(ax + b) dx
  6. Find the integral: ∫ √(ax + b) dx
  7. Find the integral: ∫ x √(x + 2) dx
  8. Find the integral: ∫ x √(1 + 2x²) dx
  9. Find the integral: ∫ (4x + 2) √(x² + x + 1) dx
  10. Find the integral: ∫ 1 / (x − √x) dx
  11. Find the integral: ∫ x / √(x + 4) dx (x > 0)
  12. Find the integral: ∫ (x³ − 1)^(1/3) x⁵ dx
  13. Find the integral: ∫ x² / (2 + 3x³)³ dx
  14. Find the integral: ∫ 1 / (x(log x)^m) dx, (m ≠ 1)
  15. Find the integral: ∫ x / (9 − 4x²) dx
  16. Find the integral: ∫ e²ˣ⁺³ dx
  17. Find the integral: ∫ x / e^(x²) dx
  18. Find the integral: ∫ e^(tan⁻¹ x) / (1 + x²) dx
  19. Find the integral: ∫ (e^(2x) − 1) / (e^(2x) + 1) dx
  20. Find the integral: ∫ (e^(2x) − e^(−2x)) / (e^(2x) + e^(−2x)) dx
  21. Find the integral: ∫ tan²(2x − 3) dx
  22. Find the integral: ∫ sec²(7 − 4x) dx
  23. Find the integral: ∫ sin⁻¹ x / √(1 − x²) dx
  24. Find the integral: ∫ (2 cos x − 3 sin x) / (6 cos x + 4 sin x) dx
  25. Find the integral: ∫ 1 / (cos²x (1 − tan x)²) dx
  26. Find the integral: ∫ cos√x / √x dx
  27. Find the integral: ∫ √(sin 2x) cos 2x dx
  28. Find the integral: ∫ cos x / √(1 + sin x) dx
  29. Find the integral: ∫ cot x log(sin x) dx
  30. Find the integral: ∫ sin x / (1 + cos x) dx
  31. Find the integral: ∫ sin x / (1 + cos x)² dx
  32. Find the integral: ∫ 1 / (1 + cot x) dx
  33. Find the integral: ∫ 1 / (1 − tan x) dx
  34. Find the integral: ∫ √(tan x) / (sin x cos x) dx
  35. Find the integral: ∫ (1 + log x)² / x dx
  36. Find the integral: ∫ (x + 1)(x + log x)² / x dx
  37. Find the integral: ∫ x³ sin(tan⁻¹(x⁴)) / (1 + x⁸) dx
  38. Choose the correct answer: ∫ (10x⁹ + 10ˣ log 10) / (10ˣ + x¹⁰) dx equals (A) 10ˣ − x¹⁰ + C (B) 10ˣ + x¹⁰ + C (C) (10ˣ − x¹⁰)⁻¹ + C (D) log(10ˣ + x¹⁰) + C
  39. Choose the correct answer: ∫ dx / (sin²x cos²x) equals (A) tan x + cot x + C (B) tan x − cot x + C (C) tan x · cot x + C (D) tan x − cot 2x + C

Trigonometric Identities in Integration

  1. Find the integral: ∫ sin²(2x + 5) dx
  2. Find the integral: ∫ sin 3x cos 4x dx
  3. Find the integral: ∫ cos 2x cos 4x cos 6x dx
  4. Find the integral: ∫ sin³(2x + 1) dx
  5. Find the integral: ∫ sin³x cos³x dx
  6. Find the integral: ∫ sin x sin 2x sin 3x dx
  7. Find the integral: ∫ sin 4x sin 8x dx
  8. Find the integral: ∫ (1 − cos x) / (1 + cos x) dx
  9. Find the integral: ∫ cos x / (1 + cos x) dx
  10. Find the integral: ∫ sin⁴x dx
  11. Find the integral: ∫ cos⁴ 2x dx
  12. Find the integral: ∫ sin²x / (1 + cos x) dx
  13. Find the integral: ∫ (cos 2x − cos 2α) / (cos x − cos α) dx
  14. Find the integral: ∫ (cos x − sin x) / (1 + sin 2x) dx
  15. Find the integral: ∫ tan³ 2x sec 2x dx
  16. Find the integral: ∫ tan⁴ x dx
  17. Find the integral: ∫ (sin³x + cos³x) / (sin²x cos²x) dx
  18. Find the integral: ∫ cos 2x + 2sin²x / cos²x dx
  19. Find the integral: ∫ 1 / (sinx cos³x) dx
  20. Find the integral: ∫ (cos 2x) / (cos x + sin x)² dx
  21. Find the integral: ∫ sin⁻¹(cos x) dx
  22. Find the integral: ∫ 1 / (cos(x − a) cos(x − b)) dx
  23. Choose the correct answer: ∫ (sin²x − cos²x) / (sin²x cos²x) dx equals (A) tan x + cot x + C (B) tan x + cosec x + C (C) −tan x + cot x + C (D) tan x + sec x + C
  24. Choose the correct answer: ∫ eˣ(1 + x) / cos²(eˣ x) dx equals (A) −cot(x eˣ) + C (B) tan(x eˣ) + C (C) tan(eˣ) + C (D) cot(eˣ) + C

Integrals of Particular Functions

  1. Find the integral: ∫ 3x² / (x⁶ + 1) dx
  2. Find the integral: ∫ 1 / √(1 + 4x²) dx
  3. Find the integral: ∫ 1 / √((2 − x)² + 1) dx
  4. Find the integral: ∫ 1 / √(9 − 25x²) dx
  5. Find the integral: ∫ 3x / (1 + 2x⁴) dx
  6. Find the integral: ∫ x² / (1 − x⁶) dx
  7. Find the integral: ∫ (x − 1) / √(x² − 1) dx
  8. Find the integral: ∫ x² / √(x⁶ + a⁶) dx
  9. Find the integral: ∫ sec²x / √(tan²x + 4) dx
  10. Find the integral: ∫ 1 / √(x² + 2x + 2) dx
  11. Find the integral: ∫ 1 / (9x² + 6x + 5) dx
  12. Find the integral: ∫ 1 / √(7 − 6x − x²) dx
  13. Find the integral: ∫ 1 / √((x − 1)(x − 2)) dx
  14. Find the integral: ∫ 1 / √(8 + 3x − x²) dx
  15. Find the integral: ∫ 1 / √((x − a)(x − b)) dx
  16. Find the integral: ∫ (4x + 1) / √(2x² + x − 3) dx
  17. Find the integral: ∫ (x + 2) / √(x² − 1) dx
  18. Find the integral: ∫ (5x − 2) / (1 + 2x + 3x²) dx
  19. Find the integral: ∫ (6x + 7) / √((x − 5)(x − 4)) dx
  20. Find the integral: ∫ (x + 2) / √(4x − x²) dx
  21. Find the integral: ∫ (x + 2) / √(x² + 2x + 3) dx
  22. Find the integral: ∫ (x + 3) / (x² − 2x − 5) dx
  23. Find the integral: ∫ (5x + 3) / √(x² + 4x + 10) dx
  24. Choose the correct answer: ∫ dx / (x² + 2x + 2) equals (A) x tan⁻¹(x + 1) + C (B) tan⁻¹(x + 1) + C (C) (x + 1) tan⁻¹x + C (D) tan⁻¹x + C
  25. Choose the correct answer: ∫ dx / √(9x − 4x²) equals (A) (1/9) sin⁻¹((9x − 8)/8) + C (B) (1/2) sin⁻¹((8x − 9)/9) + C (C) (1/3) sin⁻¹((9x − 8)/8) + C (D) (1/2) sin⁻¹((9x − 8)/8) + C

Partial Fractions

  1. Find the integral: ∫ x/((x+1)(x+2)) dx
  2. Find the integral: ∫ 1/(x²−9) dx
  3. Find the integral: ∫ 3x−1/((x−1)(x−2)(x−3)) dx
  4. Find the integral: ∫ x/((x−1)(x−2)(x−3)) dx
  5. Find the integral: ∫ 2x/(x²+3x+2) dx
  6. Find the integral: ∫ 1−x²/(x(1−2x)) dx
  7. Find the integral: ∫ x/((x²+1)(x−1)) dx
  8. Find the integral: ∫ x/((x−1)²(x+2)) dx
  9. Find the integral: ∫ 3x+5/(x³−x²−x+1) dx
  10. Find the integral: ∫ (2x−3)/((x²−1)(2x+3)) dx
  11. Find the integral: ∫ 5x/((x+1)(x²−4)) dx
  12. Find the integral: ∫ (x³+x+1)/(x²−1) dx
  13. Find the integral: ∫ 2/(1−x)(1+x²) dx
  14. Find the integral: ∫ 3x−1/(x+2)² dx
  15. Find the integral: ∫ 1/(x⁴−1) dx
  16. Find the integral: ∫ 1/(x(xⁿ+1)) dx
  17. Find the integral: ∫ cos x/((1−sin x)(2−sin x)) dx
  18. Find the integral: ∫ (x²+1)(x²+2)/((x²+3)(x²+4)) dx
  19. Find the integral: ∫ 2x/((x²+1)(x²+3)) dx
  20. Find the integral: ∫ 1/(x(x⁴−1)) dx
  21. Find the integral: ∫ 1/(eˣ−1) dx [Hint: put eˣ=t]
  22. Choose the correct answer: ∫ x dx/((x−1)(x−2)) equals (A) log|(x−1)²/(x−2)|+C (B) log|(x−2)²/(x−1)|+C (C) log|(x−1)/(x−2)²|+C (D) log|(x−1)(x−2)|+C
  23. Choose the correct answer: ∫ dx/(x(x²+1)) equals (A) log|x|−(1/2)log(x²+1)+C (B) log|x|+(1/2)log(x²+1)+C (C) −log|x|+(1/2)log(x²+1)+C (D) (1/2)log|x|+log(x²+1)+C

Integration by Parts

  1. Find the integral: ∫ x sin x dx
  2. Find the integral: ∫ x sin 3x dx
  3. Find the integral: ∫ x² eˣ dx
  4. Find the integral: ∫ x logx dx
  5. Find the integral: ∫ x log 2x dx
  6. Find the integral: ∫ x² log x dx
  7. Find the integral: ∫ x sin⁻¹x dx
  8. Find the integral: ∫ x tan⁻¹x dx
  9. Find the integral: ∫ x cos⁻¹x dx
  10. Find the integral: ∫ (sin⁻¹x)² dx
  11. Find the integral: ∫ x cos⁻¹x/√(1−x²) dx
  12. Find the integral: ∫ x sec²x dx
  13. Find the integral: ∫ tan⁻¹x dx
  14. Find the integral: ∫ x·(logx)² dx
  15. Find the integral: ∫ (x²+1)logx dx
  16. Find the integral: ∫ eˣ(sinx + cosx) dx
  17. Find the integral: ∫ x eˣ / (1 + x)² dx
  18. Find the integral: ∫ eˣ(1+sinx)/(1+cosx) dx
  19. Find the integral: ∫ eˣ (1/x − 1/x²) dx
  20. Find the integral: ∫ (x − 3) eˣ / (x − 1)³ dx
  21. Find the integral: ∫ e^(2x) sin x dx
  22. Find the integral: ∫ sin⁻¹(2x/(1 + x²)) dx
  23. Choose the correct answer: ∫ x² eˣ³ dx equals (A) (1/3)eˣ³+C (B) (1/3)eˣ²+C (C) (1/2)eˣ³+C (D) (1/2)eˣ²+C
  24. Choose the correct answer: ∫ eˣ sec x(1+tanx) dx equals (A) eˣ cosx+C (B) eˣ secx+C (C) eˣ sinx+C (D) eˣ tanx+C

Special Integrals (√(a²−x²), √(x²±a²))

  1. Find the integral: ∫ √(4−x²) dx
  2. Find the integral: ∫ √(1−4x²) dx
  3. Find the integral: ∫ √(x²+4x+6) dx
  4. Find the integral: ∫ √(x²+4x+1) dx
  5. Find the integral: ∫ √(1−4x−x²) dx
  6. Find the integral: ∫ √(x²+4x−5) dx
  7. Find the integral: ∫ √(1+3x−x²) dx
  8. Find the integral: ∫ √(x²+3x) dx
  9. Find the integral: ∫ √(1+x²/9) dx
  10. Choose the correct answer: ∫ √(1+x²) dx is equal to (A) (x/2)√(1+x²)+(1/2)log|x+√(1+x²)|+C (B) (2/3)(1+x²)^(3/2)+C (C) (2/3)x(1+x²)^(3/2)+C (D) (x²/2)√(1+x²)+(1/2)x²log|x+√(1+x²)|+C
  11. Choose the correct answer: ∫ √(x²−8x+7) dx is equal to (A) (1/2)(x−4)√(x²−8x+7)+9log|x−4+√(x²−8x+7)|+C (B) (1/2)(x+4)√(x²−8x+7)+9log|x+4+√(x²−8x+7)|+C (C) (1/2)(x−4)√(x²−8x+7)−3√2 log|x−4+√(x²−8x+7)|+C (D) (1/2)(x−4)√(x²−8x+7)−(9/2)log|x−4+√(x²−8x+7)|+C

Definite Integrals — Second Fundamental Theorem

  1. Evaluate: ∫₋₁¹ (x+1) dx
  2. Evaluate: ∫₂³ (1/x) dx
  3. Evaluate: ∫₁² (4x³−5x²+6x+9) dx
  4. Evaluate: ∫₀^(π/4) sin 2x dx
  5. Evaluate: ∫₀^(π/2) cos 2x dx
  6. Evaluate: ∫₄⁵ eˣ dx
  7. Evaluate: ∫₀^(π/4) tan x dx
  8. Evaluate: ∫_{π/6}^{π/4} cosec x dx
  9. Evaluate: ∫₀¹ dx/√(1−x²)
  10. Evaluate: ∫₀¹ dx/(1+x²)
  11. Evaluate: ∫₂³ dx/(x²−1)
  12. Evaluate: ∫₀^(π/2) cos²x dx
  13. Evaluate: ∫₂³ x dx/(x²+1)
  14. Evaluate: ∫₀¹ (2x+3)/(5x²+1) dx
  15. Evaluate: ∫₀¹ x eˣ² dx
  16. Evaluate: ∫₁² 5x²/(x²+4x+3) dx
  17. Evaluate: ∫₀^(π/4) (2sec²x + x³ + 2) dx
  18. Evaluate: ∫₀^π (sin²(x/2) − cos²(x/2)) dx
  19. Evaluate: ∫₀² (6x+3)/(x²+4) dx
  20. Evaluate: ∫₀¹ (xeˣ + sin(πx/4)) dx
  21. Choose the correct answer: ∫₀^√3 dx/(1+x²) equals (A) π/3 (B) 2π/3 (C) π/6 (D) π/12
  22. Choose the correct answer: ∫₀^(2/3) dx/(4+9x²) equals (A) π/6 (B) π/12 (C) π/24 (D) π/4

Definite Integrals by Substitution

  1. Evaluate: ∫₀¹ x/(x²+1) dx
  2. Evaluate: ∫₀^(π/2) √(sin φ) cos⁵φ dφ
  3. Evaluate: ∫₀¹ sin⁻¹(2x/(1+x²)) dx
  4. Evaluate: ∫₀² x√(x+2) dx [put x+2=t²]
  5. Evaluate: ∫₀^(π/2) sinx/(1+cos²x) dx
  6. Evaluate: ∫₀² dx/(x+4−x²)
  7. Evaluate: ∫₋₁¹ dx/(x²+2x+5)
  8. Evaluate: ∫₁² (1/x−1/(2x²)) e^(2x) dx
  9. Choose the correct answer: The value of ∫_{1/3}^{1} (x−x³)^(1/3)/x⁴ dx is (A) 6 (B) 0 (C) 3 (D) 4
  10. Choose the correct answer: If f(x) = ∫₀ˣ t sin t dt, then f'(x) is (A) cosx+x sinx (B) x sinx (C) x cosx (D) sinx+x cosx

Properties of Definite Integrals

  1. Evaluate: ∫₀^(π/2) cos²x dx
  2. Evaluate: ∫₀^(π/2) √(sinx)/(√(sinx)+√(cosx)) dx
  3. Evaluate: ∫₀^(π/2) sin^(3/2)x / (sin^(3/2)x + cos^(3/2)x) dx
  4. Evaluate: ∫₀^(π/2) cos⁵x/(sin⁵x+cos⁵x) dx
  5. Evaluate: ∫₋₅⁵ |x+2| dx
  6. Evaluate: ∫₂⁸ |x−5| dx
  7. Evaluate: ∫₀¹ x(1−x)ⁿ dx
  8. Evaluate: ∫₀^(π/4) log(1+tanx) dx
  9. Evaluate: ∫₀² x√(2−x) dx
  10. Evaluate: ∫₀^(π/2) (2 log sinx − log sin2x) dx
  11. Evaluate: ∫_{−π/2}^{π/2} sin²x dx
  12. Evaluate: ∫₀^π x/(1+sinx) dx
  13. Evaluate: ∫_{−π/2}^{π/2} sin⁷x dx
  14. Evaluate: ∫₀^(2π) cos⁵x dx
  15. Evaluate: ∫₀^(π/2) (sinx−cosx)/(1+sinxcosx) dx
  16. Evaluate: ∫₀^π log(1+cosx) dx
  17. Evaluate: ∫₀^a √x / (√x + √(a−x)) dx
  18. Evaluate: ∫₀⁴ |x−1| dx
  19. Show that ∫₀^a f(x)g(x) dx = 2∫₀^a f(x) dx if f and g satisfy f(x)=f(a−x) and g(x)+g(a−x)=4.
  20. Choose the correct answer: The value of ∫_{−π/2}^{π/2} (x³+x cosx+tan⁵x+1) dx is (A) 0 (B) 2 (C) π (D) 1
  21. Choose the correct answer: The value of ∫₀^(π/2) log((4+3sinx)/(4+3cosx)) dx is (A) 2 (B) 3/4 (C) 0 (D) −2

Miscellaneous Exercise on Chapter

  1. Integrate: 1/(x−x³)
  2. Integrate: 1/(√(x+a)+√(x+b))
  3. Integrate: 1/(x√(ax−x²)) [Hint: x=a/t]
  4. Integrate: 1/(x²(x⁴+1)^(3/4)) [Hint: divide by x⁴]
  5. Integrate: 1/(√x+x^(1/3)) [Hint: x=t⁶]
  6. Integrate: 5x/((x+1)(x²+9))
  7. Integrate: sinx/sin(x−a)
  8. Integrate: (e^(5logx)−e^(4logx))/(e^(3logx)−e^(2logx))
  9. Integrate: cosx/√(4−sin²x)
  10. Integrate: (sin⁸x−cos⁸x)/(1−2sin²xcos²x)
  11. Integrate: 1/(cos(x+a)cos(x+b))
  12. Integrate: x³/√(1−x⁸)
  13. Integrate: eˣ/((1+eˣ)(2+eˣ))
  14. Integrate: 1/((x²+1)(x²+4))
  15. Integrate: cos³x e^(log sinx) [= cos³x · sinx]
  16. Integrate: e^(3logx)(x⁴+1)⁻¹ [= x³/(x⁴+1)]
  17. Integrate: f'(ax+b)[f(ax+b)]ⁿ
  18. Integrate: 1/√(sin³x sin(x+α))
  19. Integrate: √((1−√x)/(1+√x))
  20. Integrate: ((2+sin2x)/(1+cos2x)) eˣ
  21. Integrate: (x²+x+1)/((x+1)²(x+2))
  22. Integrate: tan⁻¹√((1−x)/(1+x))
  23. Integrate: √(x²+1)[log(x²+1)−2logx]/x⁴
  24. Evaluate: ∫_{π/2}^{π} eˣ(1−sinx)/(1−cosx) dx
  25. Evaluate: ∫₀^(π/4) sinx cosx/(cos⁴x+sin⁴x) dx
  26. Evaluate: ∫₀^(π/2) cos²x/(cos²x+4sin²x) dx
  27. Evaluate: ∫_{π/6}^{π/3} (sinx+cosx)/√(sin2x) dx
  28. Evaluate: ∫₀¹ dx/(√(1+x)−√x)
  29. Evaluate: ∫₀^(π/4) (sinx+cosx)/(9+16sin2x) dx
  30. Evaluate: ∫₀^(π/2) sin2x tan⁻¹(sinx) dx
  31. Evaluate: ∫₁⁴ [|x−1|+|x−2|+|x−3|] dx
  32. Prove: ∫₁³ dx/(x²(x+1)) = 2/3+log(2/3)
  33. Prove: ∫₀¹ xeˣ dx = 1
  34. Prove: ∫₋₁¹ x¹⁷ cos⁴x dx = 0
  35. Prove: ∫₀^(π/2) sin³x dx = 2/3
  36. Prove: ∫₀^(π/4) 2tan³x dx = 1−log2
  37. Prove: ∫₀¹ sin⁻¹x dx = π/2−1
  38. Choose the correct answer: ∫ dx/(eˣ+e^(−x)) is equal to (A) tan⁻¹(eˣ)+C (B) tan⁻¹(e^(−x))+C (C) log(eˣ−e^(−x))+C (D) log(eˣ+e^(−x))+C
  39. Choose the correct answer: ∫ cos2x/(sinx+cosx)² dx is equal to (A) −1/(sinx+cosx)+C (B) log|sinx+cosx|+C (C) log|sinx−cosx|+C (D) 1/(sinx+cosx)²+C
  40. Choose the correct answer: If f(a+b−x)=f(x), then ∫_a^b x·f(x)dx is equal to (A) ((a+b)/2)∫_a^b f(b−x)dx (B) ((a+b)/2)∫_a^b f(b+x)dx (C) ((b−a)/2)∫_a^b f(x)dx (D) ((a+b)/2)∫_a^b f(x)dx
Sample

Worked examples

3 questions from Integrals, solved step by step.

Integration by Inspection

Find the integral: ∫ √x (3x² + 2x + 3) dx

Multiply each term by x^(1/2):

3x^(5/2) + 2x^(3/2) + 3x^(1/2).

∫ = 3 · 2x^(7/2)/7 + 2 · 2x^(5/2)/5 + 3 · 2x^(3/2)/3 + C

= 6x^(7/2)/7 + 4x^(5/2)/5 + 2x^(3/2) + C

Answer: (6/7)x^(7/2) + (4/5)x^(5/2) + 2x^(3/2) + C

Integration by Substitution

Find the integral: ∫ sin(ax + b) cos(ax + b) dx

Write as (1/2) sin(2ax + 2b).

Let t = 2ax + 2b, dt = 2a dx.

∫ (1/2) sin t · dt/(2a) = −cos t/(4a) + C = −cos(2ax + 2b)/(4a) + C

Answer: −cos(2ax + 2b)/(4a) + C [equivalently: sin²(ax + b)/(2a) + C]

Trigonometric Identities in Integration

Find the integral: ∫ sin 3x cos 4x dx

Use product-to-sum: 2 sin A cos B = sin(A+B) + sin(A−B).

2 sin 3x cos 4x = sin 7x + sin(−x) = sin 7x − sin x.

∫ (sin 7x − sin x)/2 dx = −cos 7x/14 + cos x/2 + C

Answer: −cos 7x/14 + cos x/2 + C

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