List of Integrals of Exponential Functions With Proofs

What Is The List Of Integrals Of Exponential Functions?

The list of integrals of exponential functions is a reference table of antiderivatives for integrands built on $e^{x}$ or $a^{x}$, alone or multiplied by powers of $x$, sines, cosines or another exponential. Its anchor entry is

$$\int e^{x},dx = e^{x} + C,$$

which holds because $e^{x}$ is the only base whose exponential function equals its own derivative. Here $C$ is the constant of integration, added to every indefinite integral.

The symbols used in every table:

This page is the catalogue. For the step-by-step lesson on the method itself, see integration of exponential functions.

How Do You Read A Table Of Exponential Integrals?

Read every row as a promise you can check: differentiate the result column and you get the integrand back. Three reading rules keep the table honest.

• Divide only by a constant. The rows for $e^{ax}$ divide by $a$ because the derivative of $ax$ is the constant $a$. When the exponent is $x^2$ or $\sqrt{x}$, there is no division rule; the row either has the matching factor outside (like $x,e^{x^2}$) or it belongs to the error-function family.

• Other bases carry $\ln a$. Rewrite $a^{x} = e^{x\ln a}$, and the constant $\ln a$ in the exponent becomes the divisor. That uses the properties of logarithms and nothing else.

• Two correct answers can look different. Antiderivatives of the same function differ by a constant, so $e^{x+3}$ and $e^{3}e^{x}$ are the same entry, and so are $\ln(1+e^{x})$ and $x + \ln(1+e^{-x})$.

What Are The Basic Integrals Of $e^x$ And $a^x$?

The basic rows all come from one move: substitute the exponent, then divide by its constant derivative. Brainly's "Evaluate the integral: $\int 2e^{-x},dx$" is a typical case; the exponent $-x$ has derivative $-1$, so $\int 2e^{-x},dx = -2e^{-x} + C$.

The last row is the master key for the next table. Whenever the derivative of the exponent sits outside as a factor, integration by substitution with $u$ equal to the exponent finishes the job in one line.

The last row explains two results that look like tricks: $\int e^{x}(x+1),dx = x,e^{x} + C$ and $\int e^{x}\left(\ln x + \frac{1}{x}\right)dx = e^{x}\ln x + C$ for $x > 0$. In each, the bracket is a function plus its own derivative.

What Are The Integrals Of Polynomials Times Exponentials?

A power of $x$ times $e^{ax}$ is integrated by integration by parts, with $u$ the power of $x$ and $dv = e^{ax},dx$. Each round lowers the power by one, which gives the reduction formula

$$\int x^{n}e^{ax},dx = \frac{x^{n}e^{ax}}{a} - \frac{n}{a}\int x^{n-1}e^{ax},dx, \qquad a \neq 0.$$

The coefficients inside each bracket are successive derivatives of the polynomial, with alternating signs and growing powers of $\frac{1}{a}$. For $x^3 e^{ax}$ the numerators are $x^3$, $3x^2$, $6x$ and $6$, which are $x^3$ and its first three derivatives.

That pattern is the tabular method (often called the DI method). List the polynomial and its derivatives down one column until you reach zero, list $e^{ax}$ and its repeated antiderivatives down the other, then multiply diagonally with alternating signs. For $x^2 e^{x}$ the columns are $x^2, 2x, 2, 0$ and $e^{x}, e^{x}, e^{x}, e^{x}$, giving $x^2e^{x} - 2x,e^{x} + 2e^{x}$.

How Do You Integrate $e^{ax}\sin(bx)$ And $e^{ax}\cos(bx)$?

You integrate by parts twice and then solve for the integral, because the second round brings the original integral back. That answers the Quora question "What is the integration method for e^(ax) sin (bx)?" and the forum request on askIITians to find "Integral e^ax sin(bx)".

Call the integral $I = \int e^{ax}\sin(bx),dx$. Parts with $u = \sin(bx)$ and $dv = e^{ax},dx$, done twice, gives

$$I = \frac{e^{ax}\sin(bx)}{a} - \frac{b}{a^2}e^{ax}\cos(bx) - \frac{b^2}{a^2}I.$$

Move the $I$ terms to one side and divide by $1 + \frac{b^2}{a^2}$:

$$I = \frac{e^{ax}\big(a\sin(bx) - b\cos(bx)\big)}{a^2 + b^2} + C.$$

A shorter route, for readers who have met Euler's formula, takes the imaginary part of $\int e^{(a+ib)x},dx = \frac{e^{(a+ib)x}}{a+ib}$. Both routes land on the same rows.

The sign pattern is the part worth memorising: the sine row has a minus between its terms and the cosine row has a plus. Differentiating the sine row once confirms it, since the $b\cos$ and $a\sin$ pieces recombine into $(a^2 + b^2)\sin(bx)$.

Which Exponential Integrals Need The Error Function?

$\int e^{-x^2},dx$ cannot be written with polynomials, roots, exponentials, logarithms or trig functions, so its antiderivative is a named special function, the error function. That is the honest answer to the Quora question "What is the value of the integral of e^{-x^2} with respect to x?":

$$\int e^{-x^2},dx = \frac{\sqrt{\pi}}{2}\operatorname{erf}(x) + C, \qquad \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{x} e^{-t^2},dt.$$

The definition matches the NIST Digital Library of Mathematical Functions, and $\operatorname{erf}(1) \approx 0.8427$. A related Quora question, "What is the integral of e^(-x)/x?", hits the second special function in this family, the exponential integral $\operatorname{Ei}$: $\int \frac{e^{-x}}{x},dx = \operatorname{Ei}(-x) + C$ for $x \neq 0$.

Here $\operatorname{erfi}(x) = -i\operatorname{erf}(ix)$ is the imaginary error function, a real-valued function for real $x$. Read down the "Elementary?" column and a rule appears that no other table spells out: odd powers of $x$ times $e^{-x^2}$ are elementary, even powers are not. An odd power leaves one spare $x$ to pair with $dx$ for the substitution $u = x^2$; an even power leaves none.

What Are The Standard Definite Integrals Of Exponential Functions?

Most named definite exponential integrals run from $0$ to $\infty$, so they are improper integrals that converge because $e^{-ax} \to 0$ faster than any power of $x$ grows. The Quora question "What is the value of the integral of e-x2 from 0 to infinity, and why?" has a clean answer: $\frac{\sqrt{\pi}}{2}$.

The why is a two-dimensional detour. Square the whole-line integral $G = \int_{-\infty}^{\infty} e^{-x^2},dx$ and write it as a double integral:

$$G^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-(x^2+y^2)},dx,dy.$$

Switch to polar coordinates, where $x^2 + y^2 = r^2$ and $dx,dy = r,dr,d\theta$. The extra $r$ is the spare factor the one-variable problem lacked:

$$G^2 = \int_{0}^{2\pi}\int_{0}^{\infty} e^{-r^2},r,dr,d\theta = 2\pi \cdot \frac{1}{2} = \pi.$$

So $G = \sqrt{\pi}$, and by symmetry the half-line value is $\frac{\sqrt{\pi}}{2} \approx 0.8862$.

The $\Gamma(n+1)$ row deserves a second look. $\int_{0}^{\infty} x^{n}e^{-x},dx$ is the gamma function, the integral that extends factorials to non-integers, and it is where the polynomial-times-exponential family meets the definite table. For more on evaluating with limits, see definite integrals.

Examples Of Integrals Of Exponential Functions

Example 1

Use the list to find $\int 5e^{3x-2},dx$.

Match the row $\int e^{ax+b},dx$ with $a = 3$ and $b = -2$:

$$\int 5e^{3x-2},dx = 5 \cdot \frac{1}{3}e^{3x-2} + C$$

$$= \frac{5}{3}e^{3x-2} + C$$

Final answer: $\frac{5}{3}e^{3x-2} + C$

Example 2

Evaluate $\int_{0}^{1} e^{x^2},dx$.

The first instinct copies the $e^{ax}$ row and divides by the derivative of the exponent, which is $2x$:

$$\text{Attempt: } \int e^{x^2},dx = \frac{e^{x^2}}{2x} + C$$

Differentiate the attempt to check it, using the quotient rule:

$$\frac{d}{dx}\left[\frac{e^{x^2}}{2x}\right] = e^{x^2} - \frac{e^{x^2}}{2x^2}$$

The extra term $-\frac{e^{x^2}}{2x^2}$ means the attempt is wrong. Dividing by the derivative of the exponent only works when that derivative is a constant. Here it is $2x$, and no spare $x$ sits outside to absorb it, so $e^{x^2}$ belongs to the error-function family:

$$\int_{0}^{1} e^{x^2},dx = \frac{\sqrt{\pi}}{2}\operatorname{erfi}(1)$$

$$\approx 0.8862 \times 1.6504 \approx 1.4627$$

A numerical method gives the same value, which is how calculators handle it.

Final answer: $\frac{\sqrt{\pi}}{2}\operatorname{erfi}(1) \approx 1.4627$

Example 3

Find $\int x^2 e^{-x},dx$.

This is the Quora question "How to solve this integration $\int x^2 e^{-x},dx$". Use the tabular method with $D$ column $x^2, 2x, 2, 0$ and $I$ column $e^{-x}, -e^{-x}, e^{-x}, -e^{-x}$:

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

$$= -e^{-x}(x^2 + 2x + 2) + C$$

Check: the derivative is $e^{-x}(x^2 + 2x + 2) - e^{-x}(2x + 2) = x^2 e^{-x}$.

Final answer: $-e^{-x}(x^2 + 2x + 2) + C$

Example 4

Find $\int e^{2x}\cos(3x),dx$.

Match the cosine row with $a = 2$ and $b = 3$, so $a^2 + b^2 = 13$:

$$\int e^{2x}\cos(3x),dx = \frac{e^{2x}\big(2\cos(3x) + 3\sin(3x)\big)}{13} + C$$

Final answer: $\frac{e^{2x}}{13}\big(2\cos(3x) + 3\sin(3x)\big) + C$

Example 5

Evaluate $\int_{0}^{\infty} x^3 e^{-2x},dx$.

Substitute $t = 2x$, so $x = \frac{t}{2}$ and $dx = \frac{dt}{2}$:

$$\int_{0}^{\infty} x^3 e^{-2x},dx = \int_{0}^{\infty} \frac{t^3}{8}e^{-t},\frac{dt}{2}$$

$$= \frac{1}{16}\int_{0}^{\infty} t^3 e^{-t},dt = \frac{1}{16} \cdot \Gamma(4)$$

$$= \frac{6}{16} = \frac{3}{8}$$

Final answer: $\frac{3}{8} = 0.375$

Example 6

A drug's blood concentration after a dose is $c(t) = 20e^{-0.3t}$ mg/L, with $t$ in hours. Find the area under the concentration curve from $t = 0$ to $t = 10$, and the total area to infinity.

Use the $e^{ax}$ row with $a = -0.3$:

$$\int_{0}^{10} 20e^{-0.3t},dt = \left[-\frac{20}{0.3}e^{-0.3t}\right]_{0}^{10}$$

$$= \frac{200}{3}\left(1 - e^{-3}\right) \approx 63.3475$$

As the upper limit grows without bound, $e^{-0.3t} \to 0$, so the total area is $\frac{200}{3} \approx 66.6667$.

Final answer: about $63.3475$ mg·h/L over 10 hours, and $\frac{200}{3} \approx 66.6667$ mg·h/L in total

Why Does A Separate List For Exponential Integrals Exist?

Exponential integrals get their own list because $e^{x}$ survives integration unchanged, so every entry is "the same exponential, plus bookkeeping". The bookkeeping is what the list records.

• Constants to divide by. $e^{ax}$ needs $\frac{1}{a}$ and $a^{x}$ needs $\frac{1}{\ln a}$; nothing else changes.

• Polynomials to peel off. Each power of $x$ in front costs one round of parts, which is why the polynomial rows grow by one term per power.

• A visible boundary. $x,e^{-x^2}$ integrates in one line while $e^{-x^2}$ needs a new function. The list shows exactly where closed forms stop, and that boundary is the bell curve at the centre of statistics, whose areas are tabulated through $\operatorname{erf}$ (the Gaussian integral is the reason the normal curve's total area works out).

Where Do Exponential Integrals Show Up In Real Life?

Exponential integrals appear wherever a quantity decays or grows at a rate proportional to its size, because the area under that curve is a total.

• Medicine. The area under a drug-concentration curve (Example 6) measures total exposure, and with first-order elimination that curve is $c_0e^{-kt}$.

• Radioactivity. The total number of decays over a time window is an integral of $N_0e^{-\lambda t}$, the same model behind exponential growth and decay.

• Electric circuits. The charge that flows while a capacitor discharges is the integral of a current $I_0e^{-t/RC}$.

• Statistics. Every normal-distribution probability is an area under $e^{-x^2/2}$, evaluated through $\operatorname{erf}$.

• Damped vibrations. Bridges, car suspensions and guitar strings settle with motion like $e^{-at}\sin(bt)$, the family in the trig table.

Where Do Students Trip Up On Integrals Of Exponential Functions?

Mistake 1: Dividing By A Non-Constant Exponent Derivative

Where it slips in: Integrating $e^{x^2}$, $e^{\sqrt{x}}$ or $e^{\sin x}$ by copying the $e^{ax}$ row.

Don't do this: Writing $\int e^{x^2},dx = \frac{e^{x^2}}{2x} + C$.

The correct way: Divide only by a constant; with no matching factor outside, the integral belongs to the error-function family or needs numerical methods.

The rusher makes this one most often, because $\int e^{3x},dx = \frac{e^{3x}}{3}$ feels like a rule about exponents rather than a rule about constants. The first instinct is always to divide by the derivative of whatever sits upstairs, so the habit worth building is to ask "is that derivative a number?" before dividing.

Mistake 2: Using The Power Rule On $a^x$

Where it slips in: Integrating $2^{x}$ or $10^{x}$ right after a page of power-rule practice.

Don't do this: Writing $\int 2^{x},dx = \frac{2^{x+1}}{x+1} + C$.

The correct way: The variable is in the exponent, not the base, so rewrite $2^{x} = e^{x\ln 2}$ and get $\frac{2^{x}}{\ln 2} + C$.

The memorizer knows both rules perfectly and still swaps them, since $x^2$ and $2^x$ use the same two symbols. Reading the integrand aloud ("two to the x" versus "x squared") separates them.

Mistake 3: Dropping The Minus Sign In $e^{-x}$

Where it slips in: Integrating $e^{-x}$, $x,e^{-x}$ or $e^{-x}\sin x$, where the inner derivative is $-1$.

Don't do this: Writing $\int e^{-x},dx = e^{-x} + C$.

The correct way: Carry the $-1$: $\int e^{-x},dx = -e^{-x} + C$. In the trig rows, substitute $a = -1$ into the formula rather than reworking signs by hand.

Mistake 4: Losing The Circular Integral

Where it slips in: The second round of parts on $e^{ax}\sin(bx)$, when the original integral reappears.

Don't do this: Starting a third round of parts, which cycles forever.

The correct way: Name the integral $I$, collect both $I$ terms on one side and divide by $1 + \frac{b^2}{a^2}$.

The sign in Mistake 3 is not cosmetic. $e^{-at}\sin(bt)$ is a vibration that dies out, while $e^{at}\sin(bt)$ is one that grows without bound. The 1940 Tacoma Narrows Bridge tore apart because wind produced a negative damping effect, so its twisting oscillation grew in amplitude instead of decaying, the real-world version of a flipped sign in the exponent.

Practice Problems On The List Of Integrals Of Exponential Functions

1. Find $\int 4e^{-2x},dx$.

2. Find $\int 5^{x},dx$.

3. Find $\int x,e^{3x},dx$.

4. Find $\int e^{-x}\cos(2x),dx$.

5. Evaluate $\int_{0}^{\infty} e^{-4x^2},dx$.

6. Evaluate $\int_{0}^{\infty} x^2 e^{-3x},dx$.

Answer to Question 1: $-2e^{-2x} + C$

Answer to Question 2: $\frac{5^{x}}{\ln 5} + C$

Answer to Question 3: $e^{3x}\left(\frac{x}{3} - \frac{1}{9}\right) + C$

Answer to Question 4: $\frac{e^{-x}}{5}\big(2\sin(2x) - \cos(2x)\big) + C$

Answer to Question 5: $\frac{\sqrt{\pi}}{4} \approx 0.4431$

Answer to Question 6: $\frac{2}{27} \approx 0.0741$

The Short Version

• The list of integrals of exponential functions grows from $\int e^{x},dx = e^{x} + C$.

• $e^{ax}$ divides by $a$, $a^{x}$ divides by $\ln a$, and the division works only when the exponent's derivative is a constant.

• Polynomials times exponentials use repeated parts, best organised with the tabular method.

• $e^{ax}\sin(bx)$ and $e^{ax}\cos(bx)$ need parts twice and a solve for the repeated integral.

• $e^{-x^2}$ has no elementary antiderivative, yet its whole-line integral is exactly $\sqrt{\pi}$.

Your Next Move

Work through practice problems 1 to 6; if you divide by something that contains $x$, return to "How Do You Read A Table Of Exponential Integrals?" and check whether the exponent's derivative is a constant. Want a live Bhanzu trainer to help you build table-reading speed? Book a free demo class.

Read More

• List of integrals for every function family in one table

• Table of integrals for the standard trig, inverse-trig and quadratic forms

• Basic integration formulas for the power and reciprocal rules beside the exponential ones

• Derivatives of exponential functions for checking any row by differentiating back

• Methods of integration for choosing between substitution, parts and partial fractions