Calculate Integral of Sinxcosx

The integral of sin(x)cos(x) is a common calculus problem that appears in physics, engineering, and mathematics. This guide provides a step-by-step explanation, an interactive calculator, and practical applications.

How to Calculate the Integral of sin(x)cos(x)

To find the integral of sin(x)cos(x), we can use a trigonometric identity to simplify the expression before integrating. Here's the step-by-step process:

Recall the double-angle identity: sin(2x) = 2sin(x)cos(x)
Rearrange the identity to solve for sin(x)cos(x): sin(x)cos(x) = (1/2)sin(2x)
Integrate both sides with respect to x: ∫sin(x)cos(x)dx = (1/2)∫sin(2x)dx
Use the substitution method for the right side: let u = 2x, du = 2dx, so dx = (1/2)du
Substitute into the integral: (1/2)∫sin(u)(1/2)du = (1/4)∫sin(u)du
Integrate sin(u): (1/4)(-cos(u)) + C = (1/4)(-cos(2x)) + C
Simplify the result: -1/4 cos(2x) + C

Note: The constant of integration C is added to represent the family of antiderivatives. The negative sign in the final result comes from the integration of sin(u).

The Formula

The integral of sin(x)cos(x) can be expressed using the following formula:

∫ sin(x)cos(x) dx = -1/4 cos(2x) + C

Where:

∫ represents the integral
sin(x)cos(x) is the integrand
dx indicates integration with respect to x
C is the constant of integration

Worked Example

Let's calculate the definite integral of sin(x)cos(x) from 0 to π/2:

Apply the antiderivative formula: ∫[0 to π/2] sin(x)cos(x) dx = [-1/4 cos(2x)] evaluated from 0 to π/2
Calculate at the upper limit (π/2): -1/4 cos(π) = -1/4 (-1) = 1/4
Calculate at the lower limit (0): -1/4 cos(0) = -1/4 (1) = -1/4
Subtract the lower limit result from the upper limit result: 1/4 - (-1/4) = 1/4 + 1/4 = 1/2

The definite integral from 0 to π/2 is 1/2. This result makes sense because the area under the curve sin(x)cos(x) over this interval is positive and finite.

Applications

The integral of sin(x)cos(x) appears in various fields:

Physics: Calculating work done by periodic forces
Engineering: Analyzing AC circuits with reactive components
Mathematics: Solving differential equations involving trigonometric functions
Signal processing: Analyzing amplitude-modulated signals

Understanding this integral helps in modeling and solving real-world problems involving oscillating systems and periodic phenomena.

Frequently Asked Questions

Why is the integral of sin(x)cos(x) important?: The integral appears in many physical systems where periodic forces or oscillations are involved. It helps in calculating work, energy, and other physical quantities in such systems.
Can I use this formula for any limits of integration?: Yes, the antiderivative formula -1/4 cos(2x) + C is valid for any real numbers x. You can use it to calculate definite integrals with any limits by evaluating the antiderivative at those limits.
What if I need to integrate sin(x)cos(x) with a coefficient?: If you have a coefficient like k sin(x)cos(x), you can factor it out: k ∫ sin(x)cos(x) dx = k (-1/4 cos(2x)) + C. The coefficient affects the amplitude of the result but not the form of the antiderivative.
Is there a simpler way to remember this integral?: Yes, you can remember the double-angle identity sin(2x) = 2sin(x)cos(x), which directly relates to this integral. This identity is fundamental in trigonometric calculus.
Where can I find more advanced applications of this integral?: Advanced applications can be found in textbooks on differential equations, physics, and engineering. Many problems in these fields involve integrals of products of trigonometric functions.

About this calculator

Written by Calculator Editorial TeamPractical calculator research and UX writing

Reviewed by Practical Tools ReviewFormula logic, assumptions, and usability checks

Updated June 25, 2026. Formulas, assumptions, and limitations are shown directly on this page.

Formula and Source

The formula used in this calculator is based on standard trigonometric identities and integration techniques. The antiderivative was derived using substitution and the double-angle identity.

For more information, you can refer to calculus textbooks or online resources on trigonometric integrals.