Problem 44
Question
Integrals involving tan \(x\) and sec \(x\) Evaluate the following integrals. $$\int_{0}^{\pi / 4} \tan ^{3} \theta \sec ^{2} \theta d \theta$$
Step-by-Step Solution
Verified Answer
To evaluate the given definite integral, we performed a substitution (u = tan θ) and found the new bounds of integration accordingly. The original integral was then transformed into a simpler integral (∫u^3 du), which we evaluated to obtain the result, 1/4.
1Step 1: Substitution
We can simplify the integrand by performing a substitution. Let
$$u = \tan \theta \Rightarrow du = \sec^2 \theta d\theta$$
Now, we need to change the bounds of integration accordingly.
When \(\theta = 0\), \(u = \tan(0)=0\). When \(\theta = \frac{\pi}{4}\), \(u = \tan(\frac{\pi}{4})=1\). Thus, the integral now becomes:
$$\int_{0}^{1} u^3 du$$
2Step 2: Evaluate the integral
Now, we can evaluate the definite integral:
$$\int_{0}^{1} u^3 du = \frac{u^4}{4} \Big|_0^1$$
3Step 3: Find the result
Substitute the bounds of integration:
$$\frac{u^4}{4}\Big|_0^1 = \frac{1^4}{4}-\frac{0^4}{4}=\frac{1}{4}-0=\frac{1}{4}$$
So, the value of the definite integral is \(\frac{1}{4}\).
Key Concepts
Definite IntegralU-substitutionIntegration BoundsTrigonometric Integrals
Definite Integral
Understanding the definite integral is essential for comprehending how to compute the area under a curve between two points. It is the fundamental process of integrating a function between two specified limits, often represented by the lower and upper bounds. In our exercise, the integral is defined from \(0\) to \(\frac{\pi}{4}\) with respect to \(\theta\).
The evaluation of a definite integral results in a real number, which provides the net area between the function and the \(x\)-axis. In the context of this problem, after the necessary u-substitution, the definite integral of \(u^3\) from \(0\) to \(1\) is evaluated to find the specific area under the curve \(u^3\) between these two points. The result is obtained by subtracting the integrated function calculated at the lower bound from that at the upper bound.
The evaluation of a definite integral results in a real number, which provides the net area between the function and the \(x\)-axis. In the context of this problem, after the necessary u-substitution, the definite integral of \(u^3\) from \(0\) to \(1\) is evaluated to find the specific area under the curve \(u^3\) between these two points. The result is obtained by subtracting the integrated function calculated at the lower bound from that at the upper bound.
U-substitution
U-substitution is a powerful technique used to simplify complex integrals. By choosing a substitution, u, for a portion of the integrand, we can often convert the integral into a simpler form that is more straightforward to evaluate. It is comparable to applying a change of variables in the integral.
In our integral involving \(\tan(x)\) and \(\sec(x)\), we let \(u = \tan(\theta)\) and accordingly \(du = \sec^2(\theta) d\theta\). This simplifies the integrand to \(u^3 du\), making it much easier to integrate. The choice of u influences the ease of integration, so identifying an appropriate u-substitution is a crucial skill in calculus.
In our integral involving \(\tan(x)\) and \(\sec(x)\), we let \(u = \tan(\theta)\) and accordingly \(du = \sec^2(\theta) d\theta\). This simplifies the integrand to \(u^3 du\), making it much easier to integrate. The choice of u influences the ease of integration, so identifying an appropriate u-substitution is a crucial skill in calculus.
Integration Bounds
In the context of definite integrals, the integration bounds define the limits between which we are finding the area under the curve. When performing a u-substitution, these bounds need to be changed to correspond to the new variable.
In our problem, we change the bounds from \(0\) and \(\frac{\pi}{4}\) in terms of \(\theta\) to \(0\) and \(1\) in terms of \(u\), as \(u\) is equivalent to \(\tan(\theta)\). These new bounds are critical because they ensure that we are evaluating the integral over the correct interval after the substitution.
In our problem, we change the bounds from \(0\) and \(\frac{\pi}{4}\) in terms of \(\theta\) to \(0\) and \(1\) in terms of \(u\), as \(u\) is equivalent to \(\tan(\theta)\). These new bounds are critical because they ensure that we are evaluating the integral over the correct interval after the substitution.
Trigonometric Integrals
Trigonometric integrals involve the integration of trigonometric functions such as sine, cosine, tangent, and secant. These types of integrals often require specific techniques of integration, including u-substitution, trigonometric identities, or integration by parts.
The initial integral in this exercise is a perfect example, featuring \(\tan^3(\theta)\) and \(\sec^2(\theta)\). The interrelation between these trigonometric functions facilitates u-substitution, since \(\tan(\theta)\) differentiated gives us \(\sec^2(\theta)\). Knowing the derivatives and integrals of basic trigonometric functions is vital for solving these types of problems.
The initial integral in this exercise is a perfect example, featuring \(\tan^3(\theta)\) and \(\sec^2(\theta)\). The interrelation between these trigonometric functions facilitates u-substitution, since \(\tan(\theta)\) differentiated gives us \(\sec^2(\theta)\). Knowing the derivatives and integrals of basic trigonometric functions is vital for solving these types of problems.
Other exercises in this chapter
Problem 44
Evaluate the following integrals. $$\int \frac{d x}{x^{3} \sqrt{x^{2}-1}}, x>1$$
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Evaluate the following integrals. $$\int \frac{x^{2}+3 x+2}{x\left(x^{2}+2 x+2\right)} d x$$
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Use integration by parts to derive the following reduction formulas. $$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \qua
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Use the approaches discussed in this section to evaluate the following integrals. $$\int_{0}^{1} \sqrt{1+\sqrt{x}} d x$$
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