Integral of Cos(E^X)

Integral of cos(e^x) along with its formula and proof with examples. Also learn how to calculate integration of cos(e^x) with step by step examples.

Alan Walker-

Published on 2023-04-14

Introduction to integral of cos(e^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 cos(e^x) integral by using different integration techniques.

What is the integral of cos (e^x)?

The integral of cos(e^x) is an antiderivative of the cosine function which is equal to sin(ex)/ ex. It is also known as the reverse derivative of the cosine function which is a trigonometric identity.

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

cos = adjacent side/hypotenuse

Integral of cos(e^x) formula

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

$\int \cos e^x dx = \frac{\sin e^x}{e^x} + c$

Or, it can be written as;

$\int \cos e^xdx = Ci(e^x) + c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. In the above integration formula, replacing the function cos(e^x) by cos(lnx) will give the integral of cos(ln x).

How to calculate the integral of cos(ex)?

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

  1. Derivatives
  2. Substitution method
  3. Definite integral

Integral of cos(ex) by using derivatives

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. Therefore, we can use the derivative to calculate the integral of a function. Let’s discuss calculating the integral of cos ex by using derivatives.

Proof of integral of cos ex by using derivatives

Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integral of cos ex by using its derivative. For this, we have to look for some derivatives formulas or a formula that gives cos x as the derivative of any function.

In derivative, we know that,

$\frac{d}{dx}(\sin e^x) = e^x \cos e^x$

It means that the derivative of sin ex gives us cos x. But it has a negative sign. Therefore, to obtain the integral of cos, we have to use it as the integral of cos, that is:

$\frac{d}{dx}(\sin e^x) = e^x \cos e^x$

Hence the integral of cos ex is equal to the negative of cos x. It is written as:

$\int \cos e^xdx = \frac{\sin e^x}{e^x} + c$

Thus, the differentiation of a function also allows us to evaluate the integral of cos(pi x).

Integral of cos ex 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 cos ex by using substitution method

To prove the integral of cos ex by using the substitution method, suppose that:

$y = \cos e^x$

Differentiating with respect to x,

$\frac{dy}{dx}= -e^x \sin e^x$

To calculate integral, we can write the above equation as:

$dy = - e^x \sin e^x.dx$

By trigonometric identities, we know that sin ex = √1 - cos²ex. Then the above equation becomes,

$dy = - e^x\sqrt{1 - cos^2e^x}.dx$

Now, substituting the value of sin2e^x, such as:

$dy = - e^x\sqrt{1 – y^2}.dx$

Multiplying both sides by cos x,

$-\frac{cos e^xdy}{e^x\sqrt{1 - y^2}} = \cos e^x.dx$

Again substitute cos x = y on the left side.

$\frac{y dy}{e^x\sqrt{1 - y^2}} = \cos e^x.dx$

Integrating on both sides by applying integral,

$-\int \frac{ydy}{e^x \sqrt{1 - y^2}} =\int \cos e^xdx$

Here we will use the u-substitution method calculator to evaluate the above expression further. For this, suppose that 1 - y² = u.


$-2y dy = du\quad \text{or}\quad y dy = -1/2 du$

Then the above left-hand side integral becomes,

$\frac{1}{2}\int \frac{1}{e^x\sqrt u}du =\int \cos e^xdx$

$\frac{1}{2}\int \frac{u^{-1/2}}{e^x} du =\int \cos e^x dx$

Since the power rule of integration calculator is

$\int x^ndx =\frac{x^{n+1}}{n+1}+C$

Therefore, by using this formula we get,

$\frac{1}{2}\frac{u^{1/2}}{e^x(1/2)} + C =\int \cos e^x dx$

$\frac{u^{1/2}}{e^x} + C =\int \cos e^x dx$

Again substituting u = 1 - y², we get

$(1 - y^2)^{\frac{1}{2}} + C =\int \cos e^x dx$

And again Substitute y = cos x here,

$(1 - cos^2 e^x)^{\frac{1}{2}} + C =\int \cos e^x dx$

$(\sin^2 e^x)^{\frac{1}{2}} + C =\int \cos e^x dx$

$\sin e^x + C =\int \cos e^x dx$

Hence the cos(e^x) integral is sin ex. Moreover, to solve the integral of a non-linear function with a square root, use our trig-substitution calculator.

Integration of cos(e^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:

$\int^b_a f(x) dx = F(b) – F(a)$

Let’s understand the verification of the integral of cos ex by using the definite integral.

Proof of integral of cos(e^x) by using definite integral

To compute the integral of cos 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 cos ex from 0 to π. For this we can write the integral as:

$\int^\pi_0 \cos e^x dx = \left|\frac{\sin e^x}{e^x}\right|^\pi_0$

Now, substitute the limit in the given function.

$\int^\pi_0 \cos e^x dx = \frac{\sin(e^π)}{e^{\pi}} - \frac{\sin (e^0)}{e^0}$

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

$\int^\pi_0 \cos e^x dx = 0$

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

$\int^{\frac{\pi}{2}}_0 \cos e^x dx = \left|\frac{\sin e^x}{e^x}\right|^{\frac{\pi}{2}}_0$


$\int^{\frac{\pi}{2}}_0 \cos e^x dx = \frac{\sin(e^{π/2})}{e^{\pi/2}} - \frac{\sin (e^0)}{e^0}$

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

$\int^{\frac{\pi}{2}}_0 \cos e^x dx= 1 - 0=1$

Therefore, the definite cos(e^x) integral is equal to 1. To avoid the above long-term calculations, our definite integration calculator provides you with an easy way to compute the definite integral of cos(ex)

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