Integration of Sin x Cos x
Integration of sin x cos x is a process of determining the integral of sin x cos x with respect to x. Integration of sin x cos x can be done using different methods of integration. Before evaluating the integral of sin x cos x, let us recall the trigonometric formula which consists of sin x cos x, which is sin 2x = 2 sin x cos x. Integration is the reverse process of differentiation, and hence the integration of sin x cos x is also called the antiderivative of sin x cos x.
In this article, we will study the integration of sin x cos x and derive its formula using the substitution method and sin 2x formula. We will also calculate the integration of sin x cos x from 0 to π.
What is Integration of Sin x Cos x?
The integration of sin x cos x gives the area under the curve of the function f(x) = sin x cos x and yields different equivalent answers when evaluated using different methods of integration. The integration of sin x cos x yields (1/4) cos 2x + C as the integral of sin x cos x using the sin 2x formula of trigonometry. Mathematically, the integral fo sin x cos x is written as ∫sin x cos x dx = (1/4) cos 2x + C, where C is the constant of integration, ∫ denotes the sign of integration and dx shows that the integration is with respect to x. Let us go through the formulas for the integration of sin x cos x.
Integration of Sin x Cos x Formula
Now, we will write the formulas for the integration of sin x cos x when evaluated using different formulas and methods of integration. Integral of sin x cos x can be determined using the sin 2x formula, and substitution method. Integration of sin x cos x given by:
 ∫ sin x cos x dx = (1/4) cos 2x + C [When evaluated using the sin 2x formula]
 ∫ sin x cos x dx = (1/2) cos^{2}x + C [When evaluated by substituting cos x]
 ∫ sin x cos x dx = (1/2) sin^{2}x + C [When evaluated by substituting sin x]
Integration of Sin x Cos x Using Sin 2x Formula
We have studied the formulas for the integration of sin x cos x. Next, we will derive the formula for the integration of sin x cos x using the sin 2x formula. We will use the following trigonometric and integration formulas:
 sin 2x = 2 sin x cos x
 ∫sin (ax) dx = (1/a) cos (ax) + C
Using the above formulas, we have
∫ sin x cos x dx = ∫(2/2) sin x cos x dx [Multiplying and dividing sin x cos x by 2]
⇒ ∫ sin x cos x dx = (1/2) ∫2 sin x cos x dx
⇒ ∫ sin x cos x dx = (1/2) ∫sin 2x dx [Using sin 2x = 2 sin x cos x]
⇒ ∫ sin x cos x dx = (1/2) (1/2) cos 2x + C
⇒ ∫ sin x cos x dx = (1/4) cos 2x + C
Hence we have derived the integral of sin x cos x using the sin 2x formula.
Integration of Sin x Cos x Using Substitution Method
Now, we will prove the integration of sin x cos x using the substitution method. We will substitute sin x and cos x separately to determine the integral of sin x cos x.
Integration of Sin x Cos x by Substituting Sin x
We will use the following formulas to determine the integral of sin x cos x:
 d(sin x)/dx = cos x
 ∫x^{n} dx = x^{n+1}/(n + 1) + C
Assume sin x = u, then we have cos x dx = du. Using the above formulas, we have
∫ sin x cos x dx = ∫udu
= u^{2}/2 + C
⇒ ∫ sin x cos x dx = (1/2) sin^{2}x + C
Hence we obtained the integration of sin x cos x by substituting sin x.
Integration of Sin x Cos x by Substituting Cos x
We will use the following formulas to determine the integral of sin x cos x:
 d(cos x)/dx = sin x
 ∫x^{n} dx = x^{n+1}/(n + 1) + C
Assume cos x = v, then we have sin x dx = dv ⇒ sin x dx = dv. Using the above formulas, we have
∫ sin x cos x dx = ∫vdv
= v^{2}/2 + C
⇒ ∫ sin x cos x dx = (1/2) cos^{2}x + C
Hence we obtained the integration of sin x cos x by substituting cos x.
Integration of Sin x Cos x From 0 to π
Now, that we have calculated the integration of sin x cos x using different methods, we will use the formula ∫ sin x cos x dx = (1/4) cos 2x + C to determine the value of the definite integral of sin x cos x from 0 to π.
\(\begin{align}\int_{0}^{\pi}\sin x \cos x&=\left [ \dfrac{1}{4}\cos 2x + C \right ]_0^\pi \\&= \left ( \dfrac{1}{4}\cos 2\pi + C \right )\left ( \dfrac{1}{4}\cos 0 + C \right )\\&=\left ( \dfrac{1}{4}+C \right )\left ( \dfrac{1}{4}+C \right )\\&=0\end{align}\)
Hence the integration of sin cos x from 0 to π is equal to 0.
Important Notes on Integration of Sin x Cos x
 ∫ sin x cos x dx = (1/4) cos 2x + C [When evaluated using the sin 2x formula]
 ∫ sin x cos x dx = (1/2) cos^{2}x + C [When evaluated by substituting cos x]
 ∫ sin x cos x dx = (1/2) sin^{2}x + C [When evaluated by substituting sin x]
 The integration of sin cos x from 0 to π is equal to 0.
Related Topics on Integration of Sin x Cos x
Integration of Sin x Cos x Examples

Example 1: Determine the integral of sin 2x using the integration of sin x cos x formula.
Solution: We know that the integration of sin x cos x is (1/4) cos 2x + C and sin 2x = 2 sin x cos x. So, we have
∫sin 2x dx = ∫2 sin x cos x dx
= 2 ∫sin x cos x dx
= 2 [(1/4) cos 2x + C]
= (1/2) cos 2x + 2C
= (1/2) cos 2x + K, where K = 2C
Answer: ∫sin 2x dx = (1/2) cos 2x + K

Example 2: Evaluate the integral of sin x + cos x.
Solution: To determine the integral of sin x + cos x, we will use the formulas d(sin x)/dx = cos x, d(cos x)/dx = sin x
∫ (sin x + cos x) dx = ∫sin x dx + ∫cos x dx
= cos x  sin x + C
Answer: ∫ (sin x + cos x) dx = cos x  sin x + C
FAQs on Integration of Sin x Cos x
What is Integration of Sin x Cos x in Trigonometry?
The integration of sin x cos x gives the area under the curve of the function f(x) = sin x cos x and yields different equivalent answers when evaluated using different methods of integration.
How to Find the Integral of Sin x Cos x?
We can derive the integral of sin x cos x formula using the substitution method and sin 2x formula.
What are the Formulas for Integration of Sin x Cos x?
Integration of sin x cos x given by:
 ∫ sin x cos x dx = (1/4) cos 2x + C [When evaluated using the sin 2x formula]
 ∫ sin x cos x dx = (1/2) cos^{2}x + C [When evaluated by substituting cos x]
 ∫ sin x cos x dx = (1/2) sin^{2}x + C [When evaluated by substituting sin x]
What is the Integration of Sin x + Cos x?
The integration of sin x + cos x is cos x  sin x + C.
Is the Integration of Sin x Cos x the Same as the Anti Derivative of Sin x Cos x?
Integration is nothing but the reverse process of differentiation, so an integral of a function is the same as its antiderivative. Hence, the integration of sin x cos x is the same as the antiderivative of sin x cos x.