Introduction to integral of sin^2x/cos 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/cos x integral by using different integration techniques.

What is the integral of sin^2x/cosx?

The integral of sin^2x/cos x is an antiderivative of sine function which is equal to ln|sec x + tan x| - sin 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/cosx formula

The formula of integral of sin contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(sin2 x/cos x)dx. In mathematical form, the integral of sin2 x/cos x is:

$∫\frac{sin^2x}{\cos x}dx = \ln| \sec x + \tan x| - \sin x + c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. In the above formula, we can find the integral of sin x/cos^2x with a little change. 

How to calculate the sin2x/cosx integration?

The integral of sin2 x/cos x is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of sine by using:

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

Integral of sin²x/cos 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 the integral of two functions combined together. Let’s discuss calculating the integral of sin²x/cos x by using integral by parts calculator with steps.

Proof of integral of sin^2x/cos x by using integration by parts

Since we know that the function sine squared x divided by cos x can be written as the product of two functions. Therefore, we can calculate the integral of sin^2x/cosx by using integration by parts. For this, suppose that:

$I = \frac{\sin^2x}{\cos x}$

Applying the integral we get,

$I = ∫\frac{\sin^2x}{\cos x}dx$

The above integral can be written as:

$I = ∫\sin x\left(\frac{\sin x}{\cos x}\right)dx$

Therefore,

$I = ∫\tan x.\sin x dx$

Since the method of integration by parts is:

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

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

$I = \tan x.∫\sin xdx - ∫[(\tan x)’.∫\sin xdx]dx$

$I = - \tan x.\cos x – ∫[\sec^2x.(-\cos x)]dx$

It can be written as:

$I = -\sin x + ∫[\sec x]dx$

Now integrating the remaining term,

$I = - \sin x + \ln|\sec x + \tan x|$

Hence the derivation of sin2x/cosx integration is,

$I = -\sin x + \ln|\sec x + \tan x|$

Integral of sin2 x/cos 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 sin by using the substitution method.

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

To proof the integral of sin2xcos x by using substitution formula, suppose that:

$I = ∫\frac{\sin^2x}{\cos x}dx$

Suppose that, the above integral can be written by using following formula,

$\sin^2x = 1 – \cos^2x$

Then,

$I = ∫\left(\frac{1 – \cos^2x}{\cos x}\right)dx$

Moreover,

$I = ∫\left(\frac{1}{\cos x} – \frac{\cos^2x}{cos x}\right)dx$

Then we get,

$I = ∫(\sec x – \cos x)dx$

Now integrating all terms.

$I = \ln|\sec x+ \tan x| -\sin x +c$

Hence the integral of sin^2x/cos x is verified.

Integral of sin2 x/cos 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 sin2 x/cos x by using the definite integral.

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

To compute the sin2x/cosx integration 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 sin2 xcos x from 0 to π. For this we can write the integral as:

$∫^π_0 \frac{\sin^2x}{\cos x}dx =\left| \ln|\sec x+ \tan x| - \sin x \right|^π_0$

Now, substituting the limit in the given function.

$∫^π_0 \frac{\sin^2x}{\cos x}dx=\ln|\sec(π) – \sec(0)| - \sin π + \sin 0$

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

$∫^π_0 \frac{\sin^2x}{\cos x}dx = \ln|1 – 1| + 0 =0$

Which is the calculation of the definite integral of sin2 x/cos x. Now to calculate the integral of sin2 x/cos x between the interval 0 to π/2, we just have to replace π by π/2. Therefore,

$∫^{\frac{π}{2}}_0 \frac{\sin^2x}{\cos x}dx = \left|\ln|\sec x+ \tan x| - \sin x\right|^{\frac{π}{2}}_0$

Now,

$∫^{\frac{π}{2}}_0 \frac{\sin^2x}{\cos x} = \ln|sec\frac{π}{2} – \tan\frac{\pi}{2}|-\ln|sec 0 + tan 0| - \sin \frac{π}{2} + \sin 0$

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

$∫^{\frac{π}{2}}_0 \frac{\sin^2x}{\cos x} = \ln(-1) – 1= -1$

Therefore, the definite integral of sin^2x/cosx is equal to -1.