Introduction to integral of sin 2x / sin x

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.

Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to a trigonometric function sine. You will also understand how to compute sin2x/sin x integral by using different integration techniques.

What is the integral of sin2x/sin x?

The integral of sin2x/sin x is an antiderivative of sine function which is equal to 2sin x + c. It is also known as the reverse derivative of sine function which is a trigonometric identity.

The sine function is the ratio of opposite side to the hypotenuse of a triangle which is written as:

Sin = opposite side / hypotenuse

Integral of sin2x/sin x formula

The formula of integral of sin2x by sin x contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(sin2x/sin x)dx. In mathematical form, the integral of sin2x/sin x is:

$∫\frac{\sin2x}{\sin x}dx = 2\sin x + c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral.

How to calculate the integral of sin(2x)/sin(x)?

The integral of sin2x/sin x is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of sin x/cos^2x by using:

1. Integration by parts
2. Substitution method
3. Definite integral

Integral of sin2x/sin x by using integration by parts

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving integral of two functions combined together. Let’s discuss calculating the integral of sin2x/sin x by using integration by parts.

Proof of integral of sin2x/sin x by using integration by parts

Since we know that the function sine and cosine squared can be written as the product of two functions. Therefore, we can calculate the integral of sin2x/sin x by using integration by parts. For this, suppose that:

$I=\frac{\sin2x}{\sin x}$

Applying the integral we get,

$I = ∫\frac{\sin2x}{\sin x}dx$

Since sin2x = 2sin x.cos x, then the above integral can also be written as:

$I = ∫\frac{2\sin x.\cos x}{sin x}dx$

Then,

$I = 2∫\cos xdx$

Or,

$I = ∫(2. \cos x)dx$

Since the method of integration by parts is:

$∫[f(x).g(x)]dx = f(x).∫g(x)dx - ∫[f’(x).∫g(x)]dx$

Now replacing f(x) and g(x) by 2 and sin x, we get,

$I = 2\sin x + ∫[0.\sin x]dx$

It can be written as:

$I = 2\sin x + c$

Hence this is the derivation of integral of sin2x/sin x. To verify above calculations, we suggest you to try our integration by parts calculator online. It is a free online tool that is helpful to solve complex integrals.

Integral of sin2x/sin x by using substitution method

The substitution method involves many trigonometric formulas. We can use these formulas to verify the integrals of different trigonometric functions such as sine, cosine, tangent, etc. Let’s understand how to prove the integral of sin2x/sin x by using the u-substitution method.

Proof of Integral of sin2x/sin x by using substitution method

To proof the integral of sin2x/sin x by using substitution method, suppose that:

$y = \frac{\sin2x}{\sin x}$

In integral form,

$I = ∫\frac{\sin2x}{\sin x}.dx$

Now assume that,

$u = \sin x\quad\text{and}\quad du=\cos x dx$

Therefore, using these substitution in the integral, we get,

$I=∫\frac{\sin2x}{\sin x}.dx=∫\frac{2\sin x\cos x}{\sin x}dx$

$I = ∫2du$

Now integrating with respect to u we get,

$I = 2u + c$

Now substituting the value of u,

$I = 2\sin x + c$

Which is the integration of give integral by using substitution method.

Integral of sin2x/sin x by using definite integral

The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:

$∫^b_a f(x) dx = F(b) – F(a)$

Let’s understand the verification of the integral of sin x by using the definite integral.

Proof of integral of sin2x/sin x by using definite integral

To compute the integral of sin x by using a definite integral, we can use the interval from 0 to π or 0 to π/2. Let’s compute the integral of sin2x/sin x from 0 to π. For this we can write the integral as:

$∫^π_0 \frac{\sin2x}{\sin x}.dx = 2\sin x|^π_0$

Now, substituting the limit in the given function.

$∫^π_0 \frac{\sin2x}{\sin x}.dx = 2\sin(π) + 2\sin (0)$

Since sin 0 is equal to 0 and sin π is equal to 0, therefore,

$∫^π_0 \frac{\sin2x}{\sin x}.dx = 0$

Which is the calculation of the definite integral of sin2x/sin x. Now to calculate the integral of sin x between the interval 0 to π/2, we just have to replace π by π/2. Therefore,

$∫^π_0 \frac{\sin2x}{\sin x}.dx = 2\sin x|^{\frac{π}{2}}_0$

Now,

$∫^π_0 \frac{\sin2x}{\sin x}.dx = 2\sin (π/2) - 2\sin (0)$

Since sin 0 is equal to 0 and sin π/2 is equal to 1, therefore,

$∫^π_0 \frac{\sin2x}{\sin x}.dx = 2(1) + 0=2$

Therefore, the definite integral of sin2x/sin x is equal to 2.