Question

Verify the following indefinite integrals by differentiation. These integrals are derived in later chapters. $$\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x=2 \sin \sqrt{x}+C$$

Short answer

Question: Verify the indefinite integral \(\int \frac{\cos\sqrt{x}}{\sqrt{x}} dx = 2\sin\sqrt{x} + C\). Answer: After finding the derivative of the function \(2\sin\sqrt{x}+C\) and simplifying it, we found that the derivative matches the given integrand \(\frac{\cos \sqrt{x}}{\sqrt{x}}\). Therefore, the indefinite integral is correct: \(\int \frac{\cos\sqrt{x}}{\sqrt{x}} dx = 2\sin\sqrt{x} + C\).

Step-by-step solution

Find the derivative of \(2\sin\sqrt{x}+C\)

Using the chain rule, we need to differentiate \(2\sin\sqrt{x}\) with respect to \(x\). The chain rule states that if you have a function \(f(g(x))\), then its derivative is \((f(g(x)))' = f'(g(x))\cdot g'(x)\). Let's apply this rule to our function: $$ \frac{d(2\sin\sqrt{x})}{dx} = 2\cdot\frac{d(\sin\sqrt{x})}{d\sqrt{x}}\cdot\frac{d(\sqrt{x})}{dx} $$ Now, let's find the derivatives of \(\sin\sqrt{x}\) and \(\sqrt{x}\): $$ \frac{d(\sin\sqrt{x})}{d\sqrt{x}} = \cos\sqrt{x}, \quad \frac{d(\sqrt{x})}{dx} = \frac{1}{2\sqrt{x}} $$ Substitute these derivatives back into the chain rule equation: $$ \frac{d(2\sin\sqrt{x})}{dx} = 2\cdot\cos\sqrt{x}\cdot\frac{1}{2\sqrt{x}} $$ The derivative of the constant \(C\) is 0, so we can skip that part. Now, let's simplify the expression:

Simplify the derivative

Notice that the factor 2 from the derivative of \(\sin\sqrt{x}\) cancels out with the factor 1/2 from the derivative of \(\sqrt{x}\). Our simplified expression is: $$ \frac{d(2\sin\sqrt{x}+C)}{dx} = \frac{\cos\sqrt{x}}{\sqrt{x}} $$

Compare the integrand and simplified derivative

Now that we have the derivative, we can compare it to the integrand and verify if the given indefinite integral is correct. The integrand is given by \(\frac{\cos\sqrt{x}}{\sqrt{x}}\). We can see that the simplified derivative matches the integrand: $$ \frac{\cos\sqrt{x}}{\sqrt{x}} = \frac{\cos\sqrt{x}}{\sqrt{x}} $$ Since the derivative of the function \(2\sin\sqrt{x}+C\) matches the given integrand, we can conclude that the indefinite integral is correct. $$ \int \frac{\cos\sqrt{x}}{\sqrt{x}} dx = 2\sin\sqrt{x} + C $$

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus, particularly when dealing with composite functions. It allows us to differentiate a function that is made up of other functions. Imagine you have a function like \( f(g(x)) \). The chain rule tells us how to find the derivative of this composite function.

• First, differentiate the outer function \( f \) with respect to the inner function \( g \). This gives you \( f'(g(x)) \).
• Next, multiply this result by the derivative of the inner function \( g \) with respect to \( x \), which is \( g'(x) \).

So, the derivative of \( f(g(x)) \) would be \( f'(g(x)) \cdot g'(x) \). This "chain" of derivatives is why it’s called the chain rule. In our solution, we use this rule to compute the derivative of the expression \( 2\sin\sqrt{x} \).

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which the function's value changes as its input changes. Differentiation is crucial in calculus because it gives us valuable information about how functions behave.

• Using differentiation, we can determine slopes of curves, rates of change, and understand the behavior of a function at specific points.
• It involves rules like the power rule, product rule, quotient rule, and the chain rule, each serving different types of functions.

In the solution given, differentiation is used to find the derivative of \( 2\sin\sqrt{x} \), breaking it down using the chain rule.

Integrand

The integrand is the function that is to be integrated. In an integral expression, it is the part that appears under the integral sign \( \int \). For example, in the integral \( \int \frac{\cos\sqrt{x}}{\sqrt{x}} \, dx \), the integrand is \( \frac{\cos\sqrt{x}}{\sqrt{x}} \).

• The process of integration involves finding the antiderivative of the integrand. This helps us compute the area under the curve represented by the integrand.
• In indefinite integrals, like our example, we find a general form of the antiderivative that includes a constant \( C \), representing a family of functions.

Because the derivative of the antiderivative yields the original function (integrand), verification through differentiation confirms correctness.

Derivative

The derivative of a function measures how the function's output value changes as the input changes. It is often symbolized by \( \frac{d}{dx} \) or prime notation \( f'(x) \).

• In our exercise, we focused on finding the derivative of the function \( 2\sin\sqrt{x} + C \) to verify it matches the integrand from the original indefinite integral.
• By finding this match between the derivation and the integrand, we confirm that the function's indefinite integral is correct.

Analyzing the derivative gives insights into the function's increasing or decreasing nature, helps in plotting graphs, and solves real-world rate-related problems.