Integral Calculation - Exercise 14

We’re asked to evaluate the following indefinite integral:

$$ \int \frac{e^{\sqrt{x}} \cdot \cos(e^{\sqrt{x}})}{\sqrt{x}} \ dx $$

To simplify the integrand, we start with a substitution:

$$ t = \sqrt{x} $$

Next, we compute the differential of both sides:

$$ dt = \frac{1}{2 \cdot \sqrt{x}} \ dx $$

To match this form, we multiply and divide the original integrand by 2:

$$ \int \frac{e^{\sqrt{x}} \cdot \cos(e^{\sqrt{x}})}{\sqrt{x}} \cdot \frac{2}{2} \ dx $$

$$ = 2 \cdot \int \frac{e^{\sqrt{x}} \cdot \cos(e^{\sqrt{x}})}{2 \cdot \sqrt{x}} \ dx $$

Now we can replace \( \frac{1}{2 \cdot \sqrt{x}} \ dx \) with \( dt \):

$$ = 2 \cdot \int e^{\sqrt{x}} \cdot \cos(e^{\sqrt{x}}) \ dt $$

Substituting \( t = \sqrt{x} \), the integral becomes:

$$ 2 \cdot \int e^t \cdot \cos(e^t) \ dt $$

We now perform a second substitution to simplify further:

$$ u = e^t $$

Differentiating both sides:

$$ du = e^t \ dt $$

This allows us to rewrite the integral as:

$$ 2 \cdot \int \cos(e^t) \cdot e^t \ dt = 2 \cdot \int \cos(u) \ du $$

Now we integrate:

$$ 2 \cdot \sin(u) + c $$

Substituting back \( u = e^t \):

$$ 2 \cdot \sin(e^t) + c $$

And finally, recalling that \( t = \sqrt{x} \):

$$ 2 \cdot \sin(e^{\sqrt{x}}) + c $$

This is the solution to the integral.

And so on...