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Chapter 4: Problem 56

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(22 x^{10}-24 e^{12 x}\right) d x$$

Short Answer

Expert verified
Based on the given step-by-step solution, provide a short answer as follows: The indefinite integral of the function \(22x^{10} - 24e^{12x}\) with respect to \(x\) is given by \(2x^{11} - 2e^{12x} + C\), where \(C\) is the constant of integration.

Step by step solution

01

Identify the components and their respective integration rules

The given function consists of two parts: a polynomial term \((22x^{10})\) and an exponential term \((-24e^{12x})\). We will use the power rule for integration for the polynomial part, and the integration of exponential functions for the exponential part. Step 2: Integrate the polynomial term
02

Integrate the polynomial term

To integrate the polynomial term \((22x^{10})\) with respect to \(x\), we can use the power rule for integration, which states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\). In our case, we have \(n=10\), so the integral becomes: $$\int 22x^{10} dx = 22 \cdot \frac{x^{10+1}}{10+1} + C_1 = 2x^{11} + C_1$$ Step 3: Integrate the exponential term
03

Integrate the exponential term

To integrate the exponential term \((-24e^{12x})\) with respect to \(x\), we need to remember that the integral of \(e^{ax} dx\) is \(\frac{e^{ax}}{a} + C\). In our case, we have \(a=12\), so the integral becomes: $$\int -24e^{12x} dx = -24 \cdot \frac{e^{12x}}{12} + C_2 = -2e^{12x} + C_2$$ Step 4: Combine the integrals
04

Combine the integrals

As both terms of the function are integrated, we can sum the integrals and combine the constants into one. The indefinite integral of the entire function is given by: $$\int(22x^{10} - 24e^{12x}) dx = (2x^{11} + C_1) + (-2e^{12x} + C_2) = 2x^{11} - 2e^{12x} + C$$ where \(C = C_1 + C_2\) is the combined constant of integration. Step 5: Check your work by differentiation
05

Check your work by differentiation

Now we verify our result by taking the derivative of the integrated function with respect to \(x\): $$\frac{d}{dx}(2x^{11} - 2e^{12x} + C) = 22x^{10} - 24e^{12x}$$ As the derivative of the integrated function matches the original function, our integration is correct.

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