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2 <p>Last updated on<strong>September 30, 2025</strong></p>

3 <p>Integration is one of the fundamental concepts of calculus. It is the inverse operation of differentiation. The symbol used to denote integral is ∫. In this topic, we will learn about the integrals of sin 2x, methods to find them, tips and tricks, and examples.</p>

4 <h2>What is the Integral of Sin 2x</h2>

5 <p>The integral<a>of</a>sin 2x is -(cos 2x) / 2 + C, and C here is the<a>constant</a>of integration. It can be written as ∫sin 2x dx. There are different methods to solve the integral of sin 2x. Let’s learn the different methods to solve integration. </p>

6 <h2>Methods to Solve the Integral of Sin 2x</h2>

7 <p>To find the value of ∫sin 2x, we use the<a>substitution method</a>. The substitution method is the technique used to simplify the integral by changing the independent<a>variable</a>.</p>

8 <p>Now, let’s see how to find the value of ∫sin 2x using the substitution method.</p>

9 <p>Let’s consider 2x = u</p>

10 <p>Then 2 dx = du, that is dx = du/2</p>

11 <p>Substituting the value in ∫sin 2x,</p>

12 <p>∫sin 2x dx = ∫ sin u(du/2)</p>

13 <p>=½ ∫sin u du</p>

14 <p>Since the integral of sin x is -cos x + C, we get,</p>

15 <p>½ ∫sin u du = ½ (-cos u) + C</p>

16 <p>As u = 2x, substituting the value,</p>

17 <p>∫sin 2x dx = -(cos 2x) / 2 + C</p>

18 <h2>Tips and Tricks for Integration of Sin 2x</h2>

19 <p>Integration is considered a hard topic in<a>math</a>by students. Let’s learn a few tips and tricks to master the integration of sin 2x. </p>

20 <ul><li>To find the value of ∫sin 2x dx, let’s break it down with substitution. So let’s consider u = 2x, so du = 2dx then dx = du/2 ∫sin 2x dx = ∫sin u du/2 = -½ cos u + C As u = 2x, Then, -½ cos u + C is -½ cos 2x + C</li>

21 </ul><ul><li>The integral of sine always gives a cosine<a>term</a>and is divided by the<a>coefficient</a>of x in the sine<a>function</a>. Add a negative sign, ∫sin 2x dx = -½ cos 2x + C</li>

22 </ul><ul><li>Practice and verify the integration answer by differentiating. d/dx (-½ cos 2x) is sin 2x. </li>

23 </ul><ul><li>Use the mnemonics such as “sine integrates to negative cosine, don’t forget to divide the line!”</li>

24 </ul><h3>Explore Our Programs</h3>

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26 <h2>Common Mistakes and How to Avoid Them in the Integration of Sin 2x</h2>

25 <h2>Common Mistakes and How to Avoid Them in the Integration of Sin 2x</h2>

27 <p>Students often consider mathematics as a difficult topic, that too integration. So, to master integration, let's look at a few common mistakes and ways to avoid them.</p>

26 <p>Students often consider mathematics as a difficult topic, that too integration. So, to master integration, let's look at a few common mistakes and ways to avoid them.</p>

28 <h3>Problem 1</h3>

27 <h3>Problem 1</h3>

29 <p>Find the value of ∫sin 2x dx</p>

28 <p>Find the value of ∫sin 2x dx</p>

30 <p>Okay, lets begin</p>

29 <p>Okay, lets begin</p>

31 <p>The value of ∫sin 2x dx is -½ cos 2x + C</p>

30 <p>The value of ∫sin 2x dx is -½ cos 2x + C</p>

32 <h3>Explanation</h3>

31 <h3>Explanation</h3>

33 <p>Using the substitution method, </p>

32 <p>Using the substitution method, </p>

34 <p>Let u = 2x</p>

33 <p>Let u = 2x</p>

35 <p>Then, du = 2dx</p>

34 <p>Then, du = 2dx</p>

36 <p>So, dx = du/2 </p>

35 <p>So, dx = du/2 </p>

37 <p>∫sin 2x dx = ∫sin u du/2</p>

36 <p>∫sin 2x dx = ∫sin u du/2</p>

38 <p>= ½ ∫sin u du</p>

37 <p>= ½ ∫sin u du</p>

39 <p>As ∫sin u du = -cos u + C</p>

38 <p>As ∫sin u du = -cos u + C</p>

40 <p>½ (-cos u) + c = -½ cos u + c</p>

39 <p>½ (-cos u) + c = -½ cos u + c</p>

41 <p>As u = 2x,</p>

40 <p>As u = 2x,</p>

42 <p>-½ cos 2x + c</p>

41 <p>-½ cos 2x + c</p>

43 <p>Well explained 👍</p>

42 <p>Well explained 👍</p>

44 <h3>Problem 2</h3>

43 <h3>Problem 2</h3>

45 <p>Evaluate the definite integral of sin 2x from 0 to π/2</p>

44 <p>Evaluate the definite integral of sin 2x from 0 to π/2</p>

46 <p>Okay, lets begin</p>

45 <p>Okay, lets begin</p>

47 <p>Integral of sin 2x from 0 to π/2is 1</p>

46 <p>Integral of sin 2x from 0 to π/2is 1</p>

48 <h3>Explanation</h3>

47 <h3>Explanation</h3>

49 <p>The antiderivative of ∫sin 2x dx is -½ cos 2x + C</p>

48 <p>The antiderivative of ∫sin 2x dx is -½ cos 2x + C</p>

50 <p>Evaluate x = 0 to x = π/2 </p>

49 <p>Evaluate x = 0 to x = π/2 </p>

51 <p>[-½ cos 2x]0π/2= [-½ cos π] - [-½ cos 0]</p>

50 <p>[-½ cos 2x]0π/2= [-½ cos π] - [-½ cos 0]</p>

52 <p>Since cos π = -1 and cos 0 = 1</p>

51 <p>Since cos π = -1 and cos 0 = 1</p>

53 <p>-½ (-1) - [-½ (1)] = ½ + ½ = 1</p>

52 <p>-½ (-1) - [-½ (1)] = ½ + ½ = 1</p>

54 <p>Well explained 👍</p>

53 <p>Well explained 👍</p>

55 <h3>Problem 3</h3>

54 <h3>Problem 3</h3>

56 <p>Find the value of ∫sin(2x + 3)dx</p>

55 <p>Find the value of ∫sin(2x + 3)dx</p>

57 <p>Okay, lets begin</p>

56 <p>Okay, lets begin</p>

58 <p>The value of ∫sin(2x + 3)dx = -½ cos(2x + 3) + C</p>

57 <p>The value of ∫sin(2x + 3)dx = -½ cos(2x + 3) + C</p>

59 <h3>Explanation</h3>

58 <h3>Explanation</h3>

60 <p>Let u = 2x + 3</p>

59 <p>Let u = 2x + 3</p>

61 <p>Then du = 2dx; then dx = du/2 </p>

60 <p>Then du = 2dx; then dx = du/2 </p>

62 <p>∫sin(2x + 3)dx = ∫sin u du/2 </p>

61 <p>∫sin(2x + 3)dx = ∫sin u du/2 </p>

63 <p>= ½ ∫sin u du</p>

62 <p>= ½ ∫sin u du</p>

64 <p>With ∫sin u du = -cos u + C</p>

63 <p>With ∫sin u du = -cos u + C</p>

65 <p>½ (-cos u) + C = -½ cos u + C</p>

64 <p>½ (-cos u) + C = -½ cos u + C</p>

66 <p>Substitute back u = 2x + 3</p>

65 <p>Substitute back u = 2x + 3</p>

67 <p>= -½ cos (2x + 3) + C</p>

66 <p>= -½ cos (2x + 3) + C</p>

68 <p>Well explained 👍</p>

67 <p>Well explained 👍</p>

69 <h3>Problem 4</h3>

68 <h3>Problem 4</h3>

70 <p>Find the value of ∫sin²2x dx</p>

69 <p>Find the value of ∫sin²2x dx</p>

71 <p>Okay, lets begin</p>

70 <p>Okay, lets begin</p>

72 <p>The value of ∫sin22x dx is (x/2) - (sin 4x/8) + C</p>

71 <p>The value of ∫sin22x dx is (x/2) - (sin 4x/8) + C</p>

73 <h3>Explanation</h3>

72 <h3>Explanation</h3>

74 <p>Using the trigonometric identity</p>

73 <p>Using the trigonometric identity</p>

75 <p>Sin22x = 1 - cos 4x / 2</p>

74 <p>Sin22x = 1 - cos 4x / 2</p>

76 <p>∫sin22x dx = ∫1 - cos 4x / 2 dx </p>

75 <p>∫sin22x dx = ∫1 - cos 4x / 2 dx </p>

77 <p>= ½ ∫cos 4x dx</p>

76 <p>= ½ ∫cos 4x dx</p>

78 <p>Integrate term by term</p>

77 <p>Integrate term by term</p>

79 <p>½ ∫dx = x/2</p>

78 <p>½ ∫dx = x/2</p>

80 <p>∫cos 4x dx = sin 4x / 4</p>

79 <p>∫cos 4x dx = sin 4x / 4</p>

81 <p>So, ∫sin22x dx = x/2 - ½(sin 4x/4) + C</p>

80 <p>So, ∫sin22x dx = x/2 - ½(sin 4x/4) + C</p>

82 <p>= x/2 - sin 4x/8 + C</p>

81 <p>= x/2 - sin 4x/8 + C</p>

83 <p>Well explained 👍</p>

82 <p>Well explained 👍</p>

84 <h3>Problem 5</h3>

83 <h3>Problem 5</h3>

85 <p>Find the value of ∫sin 2x cos 2x dx</p>

84 <p>Find the value of ∫sin 2x cos 2x dx</p>

86 <p>Okay, lets begin</p>

85 <p>Okay, lets begin</p>

87 <p>The value of ∫sin 2x cos 2x dx = - cos 4x / 8 + C</p>

86 <p>The value of ∫sin 2x cos 2x dx = - cos 4x / 8 + C</p>

88 <h3>Explanation</h3>

87 <h3>Explanation</h3>

89 <p>The double-angle identity for sine:</p>

88 <p>The double-angle identity for sine:</p>

90 <p>Sin 2x cos 2x = ½ sin4x</p>

89 <p>Sin 2x cos 2x = ½ sin4x</p>

91 <p>Then, ∫sin 2x cos 2x dx = ∫1/2 sin 4x dx</p>

90 <p>Then, ∫sin 2x cos 2x dx = ∫1/2 sin 4x dx</p>

92 <p>= ½ ∫sin 4x dx</p>

91 <p>= ½ ∫sin 4x dx</p>

93 <p>Now, integrate: ∫sin 4x dx = -cos 4x / 4 + C</p>

92 <p>Now, integrate: ∫sin 4x dx = -cos 4x / 4 + C</p>

94 <p>So, ½ (-cos 4x/ 4) +C </p>

93 <p>So, ½ (-cos 4x/ 4) +C </p>

95 <p>= -cos 4x / 8 +C</p>

94 <p>= -cos 4x / 8 +C</p>

96 <p>Well explained 👍</p>

95 <p>Well explained 👍</p>

97 <h2>FAQs on Integral of Sin 2x</h2>

96 <h2>FAQs on Integral of Sin 2x</h2>

98 <h3>1.What is the integral of sin 2 x?</h3>

97 <h3>1.What is the integral of sin 2 x?</h3>

99 <p>The integral of sin 2x is -½ cos 2x + c</p>

98 <p>The integral of sin 2x is -½ cos 2x + c</p>

100 <h3>2.What is the definite integral of sin 2x from 0 to π/2</h3>

99 <h3>2.What is the definite integral of sin 2x from 0 to π/2</h3>

101 <p>We write the integral of sin 2x from 0 to π/2 as ∫ 0π/2sin 2x dx. The value of ∫ 0π/2sin 2x dx is 1</p>

100 <p>We write the integral of sin 2x from 0 to π/2 as ∫ 0π/2sin 2x dx. The value of ∫ 0π/2sin 2x dx is 1</p>

102 <h3>3.Is integral sin 2x dx the same as the integral of sin²dx?</h3>

101 <h3>3.Is integral sin 2x dx the same as the integral of sin²dx?</h3>

103 <p>No, ∫sin 2x dx and ∫sin2dx are not the same. </p>

102 <p>No, ∫sin 2x dx and ∫sin2dx are not the same. </p>

104 <h3>4.What is C in integration?</h3>

103 <h3>4.What is C in integration?</h3>

105 <p>The C in integration represents the constant of integration.</p>

104 <p>The C in integration represents the constant of integration.</p>

106 <h3>5.What are the real-life applications of integrals?</h3>

105 <h3>5.What are the real-life applications of integrals?</h3>

107 <p>Integration is used in various fields like physics, engineering, economics, biology, medicine,<a>probability</a>, and<a>statistics</a>.</p>

106 <p>Integration is used in various fields like physics, engineering, economics, biology, medicine,<a>probability</a>, and<a>statistics</a>.</p>

108 <h2>Important Glossaries for Integration of Sin 2x</h2>

107 <h2>Important Glossaries for Integration of Sin 2x</h2>

109 <ul><li><strong>Antiderivative:</strong>The result of the integration of a function is the antiderivative. Whose derivative gives the original function. For example, ∫ sin 2x = -½ cos 2x + c, here -½ cos 2x + c is the antiderivative.</li>

108 <ul><li><strong>Antiderivative:</strong>The result of the integration of a function is the antiderivative. Whose derivative gives the original function. For example, ∫ sin 2x = -½ cos 2x + c, here -½ cos 2x + c is the antiderivative.</li>

110 </ul><ul><li><strong>Substitution method:</strong>A method used to find the value of integration; here, a substitution is used to simplify the integrals. </li>

109 </ul><ul><li><strong>Substitution method:</strong>A method used to find the value of integration; here, a substitution is used to simplify the integrals. </li>

111 </ul><ul><li><strong>Constant of integration (C):</strong>The constant of integration is a constant-used integration. It represents a number that could be added to the integral function.</li>

110 </ul><ul><li><strong>Constant of integration (C):</strong>The constant of integration is a constant-used integration. It represents a number that could be added to the integral function.</li>

112 </ul><ul><li><strong>Trigonometric identity:</strong>The formula is used to simplify the trigonometric expression.</li>

111 </ul><ul><li><strong>Trigonometric identity:</strong>The formula is used to simplify the trigonometric expression.</li>

113 </ul><p>What Is Calculus? 🔢 | Easy Tricks, Limits & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math</p>

112 </ul><p>What Is Calculus? 🔢 | Easy Tricks, Limits & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math</p>

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