Problem 35
Question
Find the indefinite integral and check your result by differentiation. $$ \int \frac{1}{x^{4}} d x $$
Step-by-Step Solution
Verified Answer
The indefinite integral of \(1/x^4\) is \(-\frac{1}{3}x^{-3} + C\), where C represents the constant of integration.
1Step 1: Apply the power rule for integration
To integrate \(1/x^4\), write the integral as \(\int x^{-4} dx\) and apply the power rule. The answer is \(x^{-4+1}/(-4+1) + C\) which simplifies to \(-\frac{1}{3}x^{-3} + C\).
2Step 2: Confirmation by differentiation
Now, check the result by differentiation using the power rule: the derivative of a constant term vanishes and the derivative of \(-\frac{1}{3}x^{-3}\) is \( \frac{1}{x^4}\). This confirms the result of the integration.
Other exercises in this chapter
Problem 35
Use a symbolic integration utility to find the indefinite integral. $$ \int \frac{e^{-x}}{1+e^{-x}} d x $$
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Use formal substitution (as illustrated in Examples 5 and 6 ) to find the indefinite integral. $$ \int 12 x\left(6 x^{2}-1\right)^{3} d x $$
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Use the Trapezoidal Rule with \(n=10\) to approximate the area of the region bounded by the graphs of the equations. $$ y=x \sqrt{\frac{4-x}{4+x}}, \quad y=0, \
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Use a graphing utility to graph the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=3-2 x-x^{2}, g(x)=0 $$
View solution