Problem 23 - Essential Calculus

Question

Find the indefinite integral and check the result by differentiation. $$ \int\left(2 \sin x-5 e^{x}\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral of the function \(2 \sin x-5 e^{x}\) is \(-2\cos x - 5e^{x} + C\), where C is the constant of integration.

Step 1: Split the Integral

The integral can be split as the sum of two separate integrals. Thus, we have \( \int(2 \sin x-5 e^{x}) dx = \int2 \sin x dx - \int5e^{x}dx \)

Step 2: Apply the Basic Integration Rules

The integral of \(\sin x\) is \(-\cos x) and for \(e^{x}\) it is \(e^{x}\). Remember to apply the constants in front of these functions. Hence, the solution will be: \( -2\cos x - 5e^{x} + C\), where \( C \) is the constant of integration.

Step 3: Check Result by Differentiating

Now, differentiate the obtained result to check the solution. The derivative of \(-2\cos x\) is \(2\sin x\) and the derivative of \(-5e^{x}\) is \(-5e^{x}\). Thus, differentiating the result gets us back to the original integrand \(2 \sin x-5 e^{x}\), hence confirming that the solution is correct.

Key Concepts

Integration by Parts

Integration by parts is a powerful technique used to find the integral of products of functions. It's like the reverse process of the product rule in differentiation. The formula for integration by parts is given by:

• \[\int u \, dv = uv - \int v \, du\]

Here, \(u\) and \(dv\) are differentiable functions chosen from the original integrand. The choice of \(u\) and \(dv\) greatly affects the simplicity of the integration, so there is a common strategy to follow:

• Choose \(u\) as the function that becomes simpler when you differentiate it.
• Choose \(dv\) as the remainder of the integrand, which should be easy to integrate.

But for the given problem, integration by parts isn't necessary. The integral was split, and straightforward basic integration rules were applied, making this solution both efficient and simpler.

Basic Integration Rules

Basic integration rules are fundamental to solving indefinite integrals, like the one provided in the exercise. They simplify the process of finding antiderivatives, especially for simple functions.

• The antiderivative of \( \sin x \) is \(-\cos x + C \).
• The antiderivative of \( e^x \) is \( e^x + C \).
• Constant multipliers are simply included in the integration process.

When integrating, it's essential to remember to add the constant of integration \( C \), because integrals represent a family of functions. For our exercise, since each part of the original problem was straightforward, basic integration rules were sufficient to achieve the solution efficiently.

Differentiation Check

Differentiation check is an important step to verify the correctness of an indefinite integral. It involves deriving the obtained antiderivative and comparing it to the original integrand.

• If the differentiation of the antiderivative returns the original integrand, then the solution is correct.

In the given problem, the derived result was \[-2\cos x - 5e^x + C\].

• The derivative of \(-2\cos x\) is \(2\sin x\).
• The derivative of \(-5e^x\) is \(-5e^x\).

Putting it together, the differentiation yielded the original function \(2 \sin x - 5e^x\). This confirms that the antiderivative was accurately calculated. Always perform a differentiation check to ensure no steps were missed and mistakes were kept at bay.