{"id":8950,"date":"2024-01-30T17:06:57","date_gmt":"2024-01-30T11:36:57","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=8950"},"modified":"2024-01-30T17:07:02","modified_gmt":"2024-01-30T11:37:02","slug":"integration-of-e-power-x","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-e-power-x\/","title":{"rendered":"Integration of e power x, e power negative x, e power ax"},"content":{"rendered":"\n

The integral of the exponential function , $e^x$, can be found using basic calculus principles. The integral of $e^x$ with respect to (x) is:<\/p>\n\n\n\n

\\[
\\int e^x \\, dx = e^x + C
\\]<\/p>\n\n\n\n

Here, (C) represents the constant of integration, which is added because the process of integration determines the antiderivative up to an arbitrary constant. This result is derived from the fact that the derivative of $e^x$ is $e^x$.<\/p>\n\n\n\n

The Exponential Function: A Primer<\/h2>\n\n\n\n

The exponential function $ e^{x} $, where ( e ) is Euler’s number (approximately 2.71828), is known for its unique property of being its own derivative. This implies that the rate of change of $ e^{x} $ with respect to ( x ) is proportional to its current value, a property that makes it crucial in modeling growth processes, decay, and many natural phenomena.<\/p>\n\n\n\n

Integration of e power x by Principal of Derivatives<\/h2>\n\n\n\n

we can use the fundamental theorem of calculus, which states that if a function f is the derivative of another function F, then the integral of f is F plus a constant.<\/p>\n\n\n\n

Now we know that<\/p>\n\n\n\n

$\\frac {d}{dx} e^x = e^x$
or
$\\frac {d}{dx} (e^x+ C) = e^x$<\/p>\n\n\n\n

Where C is the constant<\/p>\n\n\n\n

Now from fundamental theorem of calculus stated above, we can say that<\/p>\n\n\n\n

\\[
\\int e^x \\, dx = e^x + C
\\]<\/p>\n\n\n\n

We can derive this using expansion series of e to the power x also<\/p>\n\n\n\n

Definite Integration of e to the power x<\/h2>\n\n\n\n

Definite integral will be represent as<\/p>\n\n\n\n

$\\int _{0}^{1} e^x \\, dx $<\/p>\n\n\n\n

To solve this, we use the antiderivative of $ e^{x} $ , which is $ e^{x} $ . So, the integral becomes:<\/p>\n\n\n\n

\\[
\\left[ e^{x} \\right]_{0}^{1}
\\]<\/p>\n\n\n\n

Now, we evaluate this expression at the upper and lower limits of the integral:<\/p>\n\n\n\n

\\[
= e^1 -e^0
\\]<\/p>\n\n\n\n


\\[
= e -1
\\]<\/p>\n\n\n\n

Integration of e power negative x<\/h2>\n\n\n\n

The integral of the exponential function , $e^{-x}$ is give as<\/p>\n\n\n\n

\\[
\\int e^{-x} \\, dx = -e^{-x} + C
\\]<\/p>\n\n\n\n

Proof<\/strong><\/p>\n\n\n\n

Let -x = t
then -dx = dt
therefore<\/p>\n\n\n\n

\\[
\\int e^{-x} \\, dx =- \\int e^{t} \\, dt = -e^{t} + C
\\]<\/p>\n\n\n\n

Substituting back,<\/p>\n\n\n\n

\\[
\\int e^{-x} \\, dx = -e^{-x} + C
\\]<\/p>\n\n\n\n

Integration of e power ax<\/h2>\n\n\n\n

The integral of ( e^{ax} ) with respect to ( x ) is straightforward due to its simple derivative. The integral is given by:<\/p>\n\n\n\n

\\[
\\int e^{ax} \\, dx = \\frac{1}{a} e^{ax} + C
\\]<\/p>\n\n\n\n

Proof<\/strong><\/p>\n\n\n\n

Let ax = t
then adx = dt
therefore<\/p>\n\n\n\n

\\[
\\int e^{ax} \\, dx =\\frac {1}{a} \\int e^{t} \\, dt = \\frac {1}{a} e^{t} + C
\\]<\/p>\n\n\n\n

Substituting back,<\/p>\n\n\n\n

\\[
\\int e^{ax} \\, dx = \\frac {1}{a} e^{ax} + C
\\]<\/p>\n\n\n\n

Special Integral based on exponential function<\/h2>\n\n\n\n

$\\int e^x( f(x) + f^{‘} (x) ) \\; dx = e^x f(x) + C$
Proof<\/strong>
$\\int e^x( f(x) + f^{‘} (x) ) dx= \\int e^x f(x) dx + \\int e^x f^{‘} (x) dx$
Now lets calculate $\\int e^x f(x) dx$ using integration by parts by taking f(x) and $e^x$ as the first function and second function
$\\int e^x f(x) dx = f(x) e^x – \\int e^x f^{‘} (x) dx$
Substituting in Above we get
$\\int e^x{ f(x) + f^{‘} (x) } dx=f(x) e^x – \\int e^x f^{‘} (x) dx + \\int e^x f^{‘} (x) dx=e^x f(x) + C $<\/p>\n\n\n\n

Solved examples<\/h2>\n\n\n\n

Question 1<\/strong>
$$
\\int e^{3x} \\, dx
$$<\/p>\n\n\n\n

Solution:<\/strong>
$$
\\int e^{3x} \\, dx = \\frac{1}{3} e^{3x} + C
$$
where $ C $ is the constant of integration.<\/p>\n\n\n\n

Question 2<\/strong><\/p>\n\n\n\n

$$
\\int e^{x} + 4 e^{2x} \\, dx
$$<\/p>\n\n\n\n

Solution:<\/strong>
$$
\\int e^{x} + 4 e^{2x} \\, dx = e^x + 2e^{2x} + C
$$
where $ C $ is the constant of integration.<\/p>\n\n\n\n

Question 3<\/strong><\/p>\n\n\n\n

A particle’s velocity $ v(t) $ increases exponentially with time as $ v(t) = e^{2t} $ m\/s. Find the displacement of the particle from time $ t = 0 $ to $ t = T $.<\/p>\n\n\n\n

Solution:<\/strong>
Displacement is the integral of velocity. So,
$$
\\text{Displacement} = \\int_0^T e^{2t} \\, dt = \\left[\\frac{1}{2} e^{2t}\\right]_0^T = \\frac{1}{2} e^{2T} – \\frac{1}{2}
$$<\/p>\n\n\n\n

Question 2<\/strong><\/p>\n\n\n\n

Solve the differential equation $ \\frac{dy}{dx} = 7e^{7x} $.<\/p>\n\n\n\n

Solution:<\/strong>
Integrating both sides with respect to $ x $,
$$
y = \\int 7e^{7x} \\, dx = 7 \\cdot \\frac{1}{7} e^{7x} + C = e^{7x} + C
$$
where $ C $ is the constant of integration.<\/p>\n\n\n\n

\n

Other Integration Related Articles<\/p>\n\n\n\n

Integration of tan inverse x<\/a><\/td>Integration of cos inverse x<\/a><\/td><\/tr>
Integration of cos square x<\/a><\/td>Integration of fractional part of x<\/a><\/td><\/tr>
integration of uv formula<\/a><\/td>integration of log x<\/a><\/td><\/tr>
integration of sin inverse x<\/a><\/td>Integration of constant term<\/a><\/td><\/tr>
integration of hyperbolic functions<\/a><\/td>List of Integration Formulas <\/a><\/td><\/tr>
integration of log sinx<\/a><\/td>integration of sin x cos x dx<\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"

The integral of the exponential function , $e^x$, can be found using basic calculus principles. The integral of $e^x$ with respect to (x) is: \\[\\int e^x \\, dx = e^x + C\\] Here, (C) represents the constant of integration, which is added because the process of integration determines the antiderivative up to an arbitrary constant. […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-8950","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nIntegration of e power x, e power negative x, e power ax - physicscatalyst's Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/integration-of-e-power-x\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of e power x, e power negative x, e power ax - physicscatalyst's Blog\" \/>\n<meta property=\"og:description\" content=\"The integral of the exponential function , $e^x$, can be found using basic calculus principles. 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