## Introduction to integral of sin x tan 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 sin tan integral by using different integration techniques.

## What is the integral of sin(x)tan(x)?

The integral of sin xtan x is an antiderivative of sine function which is equal to ln|sec x + tan x| - sin x. 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

Similarly, the function tan is the ratio of sine and cosine function. Mathematically, it is expressed as:

tan = sine / cosine

### Integral of sin x.tan x formula

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

$∫\sin x.\tan xdx = \ln|\sec x+\tan x|-\sin x + c$

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

## How to calculate the integral of sin(x)tan(x)?

The integral of sin xtan 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:

- Integration by parts
- Substitution method
- Definite integral

## Integral of sin xtan 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 sin xtan x by using integration by parts formula.

### Proof of integral of sin xtan x by using integration by parts

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

$I =\sin x.\tan x$

Applying the integral we get,

$I = ∫\sin x.\tan xdx$

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,

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

Since tan x cos x = sin x and cos x sec2x = sec x then the above equation can be written as:

$I=–\sin x+∫\sec x.dx$

Since the integral of sec x is equal to ln |sec x + tan x| then,

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

Hence we have verified the integral of sin x tan x by using integration by parts. Similarly, this method can be used to solve the integration of sin^2xcos x, because it is the product of two functions.

## Integral of sin xtan 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 x.tan x by using the substitution method.

### Proof of Integral of sin xtan x by using substitution method

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

$I=∫\sin x.\tan xdx$

The above integral can be written as:

$I=∫\sin x.\frac{\sin x}{\cos x}dx$

Or,

$I =∫\frac{\sin^2x}{\cos x}dx{2}lt;/p>

We can use trigonometric formulas in substitution method. Therefore, substituting sin^{2}x = 1 – cos2x/2, we get

$I=∫\left(\frac{1 – \cos^2x}{\cos x}\right)dx{2}lt;/p>

Then,

$I=∫\left(\frac{1}{\cos x} – \cos x\right)dx{2}lt;/p>

By integrating with respect to x we get;

$I=\ln|\sec x+\tan x|-\sin x + c{2}lt;/p>

Hence the integral of sin xtan x is equal to ln |sec x + tan x| - sin x +c.

## Integral of sin xtan 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 formula of defintie integral calculator 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 sin xtan x by using definite integral

To compute the integral of sin xtan 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 sin2 xcos x from 0 to π. For this we can write the integral as:

$∫^π_0 \sin x\tan xdx=\left|\ln|\sec x+\tan x| -\sin x\right|^π_0$

Now, substituting the limit in the given function.

$∫^π_0 \sin x\tan x dx=\ln|\sec π+\tan π |-\sin π–\ln|\sec 0+\tan 0|-\sin 0$

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

$∫^π_0 \sin x\tan xdx=\ln(-1) -1=-1$

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

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

Now,

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

therefore,

$∫^{\frac{π}{2}}_0 \sin x\tan xdx = 0 – 1= - 1$

Therefore, the definite integral of sin xtan x is equal to -1.