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Chapter 7: Problem 10

Evaluate the following integrals. $$\int e^{3-4 x} d x$$

Short Answer

Expert verified
Question: Find the integral of the function e^(3-4x) with respect to x. Answer: The integral of the function e^(3-4x) with respect to x is given by the expression $$-\frac{1}{4} e^{3-4x} + C$$, where C is the constant of integration.

Step by step solution

01

Make an appropriate substitution

Let's use substitution to solve the integral. We can let u be the exponent of the exponential function e^(3-4x), so $$u = 3 - 4x$$ Now, we will differentiate u with respect to x to find du: $$\frac{d u}{d x} = -4$$ Hence, we can find dx in terms of du: $$d x = \frac{1}{-4} d u$$
02

Replace the variables and adjust the integral

Now we can replace the variables and the differential dx in the original integral: $$\int e^{3-4 x} d x = \int e^{u} \frac{1}{-4} du$$ The integral now looks simpler and can be calculated as follows:
03

Evaluate the integral

Evaluating the simpler integral, we get: $$\int e^{u} \frac{1}{-4} d u = -\frac{1}{4} \int e^u du$$ The antiderivative of e^u with respect to u is e^u, so we have: $$-\frac{1}{4} \int e^u du = -\frac{1}{4} e^u + C$$ where C is the constant of integration.
04

#Step 4: Replace u with the original expression

Now that we have found the antiderivative, it's time to replace u with the original expression in terms of x: $$-\frac{1}{4} e^u + C = -\frac{1}{4} e^{3-4x} + C$$ This is the final answer: $$\int e^{3-4 x} d x = -\frac{1}{4} e^{3-4x} + C$$

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Most popular questions from this chapter

A differential equation of the form \(y^{\prime}(t)=F(y)\) is said to be autonomous (the function \(F\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(F\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0,\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal line segments in the direction field. Note also that for autonomous equations, the direction field is independent of \(t\). Consider the following equations. a. Find all equilibrium solutions. b. Sketch the direction field on either side of the equilibrium solutions for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=y(y-3)$$

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