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Chapter 4: Problem 107

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{1+\sqrt{x}}{x} d x$$

Short Answer

Expert verified
Question: Find the indefinite integral of the given function - $$\int (1 + \frac{1}{\sqrt{x}}) d x$$ Solution: We have found the antiderivative of the given function as: $$x + 2x\sqrt{x} + x + C$$ Now, confirm the result by taking the derivative. Derivative: $$\frac{d(x + 2x\sqrt{x} + x)}{d x} = 1 + \frac{1}{\sqrt{x}}$$ Since the derivative matches the original integrand, the indefinite integral is correct.

Step by step solution

01

Perform substitution to simplify the integral

We will use substitution to simplify the integral by setting $$u = 1 + \sqrt{x}$$ and $$x = (u - 1)^2$$. After solving for x, we will also need to find dx in terms of du.$$d x = \frac{d((u - 1)^2)}{d u} d u$$
02

Find dx in terms of du

Differentiating the equation for x with respect to u, we obtain$$d x = 2(u - 1) d u$$Now, we can substitute u and dx into the original integral.
03

Substitute u and dx into the original integral

Substituting the values of u and dx from Steps 1 and 2, we get$$\int \frac{u}{(u - 1)^2}(2 (u - 1)) d u$$which simplifies to$$\int 2u d u$$
04

Integrate the simplified integral

Now, we can integrate the simplified integral with respect to u.$$ \int 2u d u = u^2 + C$$
05

Replace u with the original variable x

Substitute the expression of u in terms of x$$u = 1 + \sqrt{x}$$$$ (1 + \sqrt{x})^2 + C = x + 2x\sqrt{x} + x$$This is now the antiderivative of the given integral.
06

Confirm the result through differentiation

Now we must check the result by taking the derivative of the antiderivative with respect to x.$$ \frac{d(x + 2x\sqrt{x} + x)}{d x} = 1 + \frac{1}{\sqrt{x}}$$This matches the initial integrand expression. Thus, our result is correct.

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