Evaluate the integral?

Question

5dx/(sq(e^x))

please give in exact form as well

thanks very much everybody.

Answer 1

I presume you mean to find:

Integral (5dx / sqrt(e^x))dx

Your form is a little confusing, but I'll solve it under those assumptions.

We can pull the constant 5 out of the integral.

5 * Integral (1/sqrt(e^x)) dx

Now, note that when taking the square root of something, it's the same as taking something to the power of 1/2.

5 * Integral (1/(e^x)^(1/2))dx

And, (e^x)^(1/2) = e^(x/2) [since we multiple the powers whenever we have a power to a power]

5 * Integral (1/[e^(x/2)])dx

We can bring the exponent to the top, and the power becomes negative.

5 * Integral (e^(-x/2))dx

And now, we solve using regular substitution.

Let u = -x/2
du = -1/2 dx
(-2) du = dx

So now we're left with

5 * Integral (e^u * (-2) du)

And as always, pull out the constants.

-10 * Integral (e^u) du

And now the integral is trivial to solve.

-10 * e^u + C

Replace u = -x/2 back, to get

-10 * e^(-x/2) + C

Remember, it is proper math etiquette to never have negative powers, so we make a denominator for e^(-x/2)

-10 * (1/e^(x/2)) + C

(-10) / e^(x/2) + C