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Chapter 13: Problem 37

Evaluate the integrals. $$ \int 100\left(1.1^{x}\right) d x $$

Short Answer

Expert verified
The short answer is: \(\int 100(1.1^x) dx = \dfrac{100}{\ln(1.1)} (1.1^x) + C\).

Step by step solution

01

Identify the base function and its derivative

The base function in the given integral is \(1.1^x\). We need to find its derivative to apply the power rule for integration. The derivative of \(a^x\) is given by the formula: \(\frac{d}{dx}(a^x) = a^x \ln(a)\). So for our base function, the derivative is: \(\frac{d}{dx} (1.1^x) = 1.1^x \ln(1.1)\).
02

Apply the power rule for integration

The power rule for integrals states that the integral of the derivative of a function multiplied by the function is equal to the function itself multiplied by a constant, i.e., \(\int f'(x) f(x) dx = f(x) + C\). We already know that the derivative of our base function is \(1.1^x \ln(1.1)\). Our given integral is \(\int 100(1.1^x) dx\). Now, multiply the given function by the derivative we found in Step 1, and then apply the power rule: \(\int 100(1.1^x) dx = \dfrac{100}{\ln(1.1)} \int 1.1^x (\ln(1.1)) dx\) \(\dfrac{100}{\ln(1.1)} \int 1.1^x (\ln(1.1)) dx = \dfrac{100}{\ln(1.1)} (1.1^x) + C\) So, the integral of \(100(1.1^x)\) with respect to x is: \(\dfrac{100}{\ln(1.1)} (1.1^x) + C\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential functions
Exponential functions form a unique family of mathematical functions where the variable appears in the exponent. An exponential function with base \(a\) can be expressed as \(f(x) = a^x\). In our example,
Integration techniques
Integration techniques are various methods used to calculate the integral of a function. The goal is to find the antiderivative, which is the opposite of differentiation.
Definite and indefinite integrals
In calculus, integrals are used to calculate areas under curves or to accumulate quantities. They can be either definite or indefinite. When answering an integration problem, knowing the

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