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How to Integrate e^sqrt(x)

The function f(x) = e^sqrt(x) is the exponential of the square root function. Because the exponent is the square root of x, rather than x, the function grows slower than the plain exponential function e^x. One solution to the separable differential equation

y' = y/(2*sqrt(x))

is y = e^sqrt(x). To integrate the function, you must use a combination of u-substitution and integration by parts. Here is the integral worked out along with an example.

Graphs of e^sqrt(x) and e^x Compared

Graphs of y = e^sqrt(x) (red) and y = e^x (blue).
Graphs of y = e^sqrt(x) (red) and y = e^x (blue).

Integrating e^sqrt(x): Step 1

The first step in solving the integral ∫ e^sqrt(x) dx is to make the substitution

x = u^2
dx = 2u du

This gives us the new integral

∫ e^u * 2u du

Now we can solve this simpler calculus problem using integration by parts.


Integrating e^sqrt(x): Step 2

To integrate e^u * 2u by parts, we assign the new variables

p = 2u
dq = e^u du

which implies that

dp = 2 du
q = e^u

Using the formula

∫ p * dq = p*q - ∫ q * dp

we get

∫ e^u * 2u du

= 2u*e^u - 2*∫ e^u du

= 2u*e^u - 2*e^u + c

Finally, using the revers substitution u = sqrt(x), we get the antiderivative of the original problem:

2*sqrt(x)*e^sqrt(x) - 2*e^sqrt(x) + c


Here is a mathematical application of integrating the exponential square root function: Solve the separable differential equation

y' = y*e^sqrt(x)

The first step in solving this calculus problem is to rewrite the equation as

dy/dx = y*e^sqrt(x)

If we algebraically separate the variables x and y, we get

(1/y) dy = e^sqrt(x) dx

Both sides can now be integrated:

∫ (1/y) dy = ∫ e^sqrt(x) dx

Ln(y) = 2*sqrt(x)e^sqrt(x) - 2*e^sqrt(x) + c

y = e^[ 2*sqrt(x)e^sqrt(x) - 2*e^sqrt(x) + c ]

The constant "c" depends on the initial conditions of the differential equation, if any are given. In this example, there are no initial conditions, so this is the most general form of the solution.


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Comments 2 comments

Gael 21 months ago

How do I integrate exp(x^1/3)?


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calculus-geometry 20 months ago from Germany Author

With the substitution x = z^3, you get (3cbrt(x^2) - 6cbrt(x) + 6)e^cbrt(x), where cbrt is the cube root.

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