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 anti-derivative 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.

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.

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) cos²x + C [When evaluated by substituting cos x]
• ∫ sin x cos x dx = (1/2) sin²x + C [When evaluated by substituting sin x]

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.

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
• ∫xⁿ dx = xⁿ⁺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

= u²/2 + C

⇒ ∫ sin x cos x dx = (1/2) sin²x + 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
• ∫xⁿ dx = xⁿ⁺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

= -v²/2 + C

⇒ ∫ sin x cos x dx = (-1/2) cos²x + C

Hence we obtained the integration of sin x cos x by substituting cos x.

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) cos²x + C [When evaluated by substituting cos x]
• ∫ sin x cos x dx = (1/2) sin²x + 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

• Integral of Tan 2x
• Arctan Integral
• Integral of Sin Inverse

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. The integral of sinx cosx is given by ∫ sin x cos x dx = (-1/4) cos 2x + C.

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) cos²x + C [When evaluated by substituting cos x]
• ∫ sin x cos x dx = (1/2) sin²x + 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 anti-derivative. Hence, the integration of sin x cos x is the same as the anti-derivative of sin x cos x.