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Magazine Chapter 8: Problem 9
Find the antiderivatives or evaluate the definite integral in each problem. $$\int \frac{\sin x}{\cos ^{3} x} d x$$
Short Answer
Expert verified
The antiderivative is \( \frac{1}{2} \sec^2 x + C \).
Step by step solution
01
Recognizing the Integral Form
Notice that the integral \( \int \frac{\sin x}{\cos^{3} x} \, dx \) can be rearranged as \( \int \sin x \cdot \cos^{-3} x \, dx \). This form can suggest the use of a substitution method.
02
Select a Substitution
Use the substitution \( u = \cos x \), which implies that \( du = -\sin x \, dx \). This substitution simplifies the integral into a form that we can integrate more easily.
03
Implement the Substitution
Substitute \( u = \cos x \) into the integral. We have \( \int \sin x \cdot \cos^{-3} x \, dx = \int \sin x \cdot u^{-3} \, dx \). Substitute \( \sin x \, dx = -du \), resulting in the integral \(-\int u^{-3} \, du \).
04
Integrate with respect to u
The integral \(-\int u^{-3} \, du \) can be solved by applying the power rule for integrals. Thus, \(-\int u^{-3} \, du = -\frac{u^{-2}}{-2} + C = \frac{1}{2}u^{-2} + C\).
05
Back Substitute the Original Variable
Replace \( u \) with \( \cos x \) to transform the expression back in terms of \( x \). Thus, the solution is \( \frac{1}{2} \cos^{-2} x + C \), which simplifies to \( \frac{1}{2} \sec^2 x + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful tool in calculus used to simplify integrals. This technique often transforms a complicated integral into a simpler one, making it easier to evaluate. In essence, you replace a portion of the integral with a variable, usually denoted as \( u \), hence the term 'u-substitution'.
Here's how it works:
Here's how it works:
- Select a substitution: Identify a function within the integral that can be replaced by \( u \). This function is often part of a composite function.
- Differentiate: Find \( du \) by differentiating your chosen substitution with respect to \( x \).
- Substitute: Replace the original function and \( dx \) with expressions in terms of \( u \) and \( du \). This transforms the integral into a new integral involving \( u \).
- Integrate: Solve the simpler integral with respect to \( u \). This step is often much easier than the original integral.
- Back substitute: After integrating, replace \( u \) with the original expression to get the solution in terms of \( x \).
Definite Integrals
Definite integrals offer a means to find the area under a curve from one point to another. These integrals provide a cumulative sum of the area and are bounded by limits of integration, say \( a \) and \( b \).
In the context of definite integrals, the fundamentals are:
In the context of definite integrals, the fundamentals are:
- The integral \( \int_a^b f(x) \, dx \) computes the net area between the function \( f(x) \) and the x-axis, from \( x=a \) to \( x=b \).
- The result might be positive, negative, or zero, depending on the function's position relative to the x-axis.
- The Fundamental Theorem of Calculus ties together differentiation and integration, allowing us to evaluate a definite integral using the antiderivative.
Power Rule for Integration
The power rule is a fundamental integration technique for finding antiderivatives. It simplifies the process when integrating functions in the form \( x^n \), where \( n \) is any real number except \(-1\).
The power rule states:
The power rule states:
- For \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), the formula applies when \( n eq -1 \).
- This rule gives us the antiderivative of a simple power function, only if \( n+1 eq 0 \).
- "C" represents the constant of integration, accounting for any vertical shifts possible in the indefinite integral.
