Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. $$\int 10^{4 t+1} d t$$

Short answer

Question: Find the indefinite integral of the function \(10^{4t + 1}\) with respect to \(t\). Answer: The indefinite integral of the function \(10^{4t + 1}\) with respect to \(t\) is \(\frac{1}{4}\frac{10^{4 t+1}}{\ln(10)}+C\), where \(C\) is the constant of integration.

Step-by-step solution

Identify the change of variable (u-substitution)

We can let the exponent \(4t + 1\) be a new variable, let's call it \(u\). So, we have \(u = 4t + 1\).

Differentiate both sides with respect to t

Differentiate \(u = 4t + 1\) with respect to \(t\) to get the relationship between dt and du: $$\frac{d u}{d t} = 4$$.

Replace dt in terms of du

Solve for dt: $$ d t=\frac{1}{4} d u $$.

Substitute and integrate

Replace the variables in the original integral with our new variables: $$ \begin{aligned} \int 10^{4 t+1} d t &= \int 10^{u} \cdot \frac{1}{4} du\\ &= \frac{1}{4}\int 10^{u} d u \end{aligned} $$

Integrate using the exponential integral formula

The integral of an exponential function is given by: $$\int a^u du = \frac{a^u}{\ln(a)} + C$$ Here, \(a = 10\), so we have: $$ \frac{1}{4}\int 10^u du = \frac{1}{4}\frac{10^u}{\ln(10)} + C $$

Substitute u back with the original variable t

Substitute \(u\) back with \(4t + 1\): $$ \frac{1}{4}\frac{10^u}{\ln(10)}+C = \frac{1}{4}\frac{10^{4 t+1}}{\ln(10)}+C $$

Check the result by differentiating

Differentiate the result with respect to \(t\): $$ \frac{d}{d t}\left(\frac{1}{4}\frac{10^{4 t+1}}{\ln(10)}+C\right) =10^{4t+1} $$ The result of the differentiation matches the original function, confirming that the integral is correct.

Key concepts

u-substitution

U-substitution is a common technique used in calculus for solving integrals, especially when dealing with composite functions. It helps simplify the integration process by introducing a new variable to replace a part of the original function. Here’s how it works:

• Identify the part of the function which can be substituted. Generally, it's the inner function of a composite function or a component that complicates the integral.
• Assign this part to a new variable, say, \(u\). For instance, in the integral \(\int 10^{4t+1} dt\), we can set \(u = 4t + 1\).
• Differentiate this equation with respect to \(t\) to find \(du\). In our example, differentiating gives \(\frac{du}{dt} = 4\).
• Solve for \(dt\) to express it in terms of \(du\). Thus, \(dt = \frac{1}{4} du\).
• Substitute \(u\) and \(dt\) back into the integral, transforming it into a simpler form that can be easily solved.

Using u-substitution helps transform complex integrals into recognizable forms, aiding in their evaluation.

exponential integral

The exponential integral involves integrating functions of the form \(a^u\), where \(a\) is a constant base raised to a variable exponent \(u\). These types of integrals are vital in calculus since exponential functions frequently appear in modeling growth, decay, and other natural phenomena.
To solve \(\int a^u \, du\), we use the following integral formula:

• \[ \int a^u \, du = \frac{a^u}{\ln(a)} + C \]
• Here, \( a\) is the base, and \(\ln(a)\) is the natural logarithm of \(a\). The result includes an arbitrary constant \(C\), representing the family of antiderivatives.

In the context of our problem, the function \(10^u\) was integrated using this formula. By integrating, and then substituting back our original variable \(t\), we were able to solve \(\int 10^{4t+1} \, dt\). This formula and method are powerful tools for quickly resolving exponential integrals, making complex integrations straightforward.

differentiation

Differentiation is the reverse process of integration and a fundamental concept in calculus. It's used to confirm if an integral result is correct. After integrating a function, differentiating the answer should yield the original function from which you started.
Here's how you use differentiation to verify an integral:

• Differentiate the resulting function of the integral with respect to the original variable.
• Check that this derivative matches the initial function inside the integral. If they match, your integration is correct.

In our exercise, after calculating the integral \(\int 10^{4t+1} \, dt\), we differentiated the result: \[ \frac{d}{dt}\left(\frac{1}{4}\frac{10^{4t+1}}{\ln(10)} + C\right) \]This differentiation reproduced the function \(10^{4t+1}\), verifying the correctness of the initial integration. Hence, differentiation not only checks for errors but also reinforces the accuracy of the calculation.