Problem 9 - Applied Calculus

Question

Find each indefinite integral. \(\int \frac{d z}{\sqrt{z}}\)

Step-by-Step Solution

Answer

\(2\sqrt{z} + C\)

Step 1: Understand the integrand

The integrand is given as \(\frac{1}{\sqrt{z}}\), which can be rewritten using exponent notation as \(z^{-1/2}\). This helps us apply the power rule for integration.

Step 2: Apply the power rule for integration

The power rule for integration states that \(\int z^n \, dz = \frac{z^{n+1}}{n+1} + C\) for \(n eq -1\). Here, \(n = -\frac{1}{2}\), so we substitute into the rule: \[\int z^{-1/2} \, dz = \frac{z^{(-1/2)+1}}{(-1/2)+1} + C = \frac{z^{1/2}}{1/2} + C\]

Step 3: Simplify the expression

Simplifying the expression \(\frac{z^{1/2}}{1/2}\) is equivalent to \(2z^{1/2}\). Therefore, the integral is \[2\sqrt{z} + C\] where \(C\) is the constant of integration.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental principle in calculus, especially when dealing with indefinite integrals. This rule helps us find the antiderivative of a function that has the form of a power of a variable. In simpler terms, if you know how to differentiate a function using the power rule, then integrating can be thought of as reversing that process.

Here's a quick breakdown of the power rule for integration:

• You apply it to expressions like \( z^n \), where \( n \) is any real number except \(-1\).
• When you integrate \( z^n \), you add 1 to the exponent (so it becomes \( n+1 \)).
• Then, you divide the expression by the new exponent \( n+1 \). This produces the antiderivative: \( \frac{z^{n+1}}{n+1} \).

Adding the constant of integration \( C \) is essential because it represents the family of all possible antiderivatives. Keep in mind, the rule changes if \( n = -1 \) as it results in a logarithmic function instead.

Exponent Notation

Exponent notation is a way of expressing numbers that involve powers, or exponents, which are numbers that tell us how many times to use the number in a multiplication. In calculus, using exponent notation can simplify the process of integrating functions.

For example, the integrand \( \frac{1}{\sqrt{z}} \) can be rewritten using exponent notation as \( z^{-1/2} \). This transforms the expression into a simpler form that is more straightforward to integrate using the power rule.

Why is this helpful? When you convert a root or fractional power into exponent notation, it aligns the function with the power rule that supplies us with a method for finding antiderivatives. Thus, exponent notation is a powerful tool that makes calculus operations more manageable and less prone to errors.

Constant of Integration

The constant of integration \( C \) is a crucial part of indefinite integrals. When you integrate a function, you're essentially finding a family of functions that could have produced the derivative you started with. This is where \( C \) comes into play.

Why do we need it? Because an indefinite integral represents all possible antiderivatives of a function. It's important to consider that when you take the derivative of a function that includes a constant, the constant disappears. For example, the derivative of \( 2\sqrt{z} + 5 \) and \( 2\sqrt{z} + 10 \) are the same: \( \frac{1}{\sqrt{z}} \).

Thus, \( C \) accounts for all possible constants that could have been present in the original function but vanished during differentiation. Anytime you see an indefinite integral, remember to include \( C \) to cover all possible solutions.