Problem 55 Integrals with $\sin ^{2} x$ a... [FREE SOLUTION]

Chapter 5: Problem 55

Integrals with $\sin ^{2} x$ and $\cos ^{2} x$ Evaluate the following integrals. $$\int \sin ^{2}\left(\theta+\frac{\pi}{6}\right) d \theta$$

Short Answer

Expert verified

Answer: The integral of $\sin^2(\theta+\frac{\pi}{6})$ with respect to $\theta$ is $\frac{1}{2}\theta - \frac{1}{4} \sin(2\theta +\frac{\pi}{3}) + C$, where $C$ is the constant of integration.

Step by step solution

Apply the double-angle formula

We begin by applying the double-angle formula to simplify the integral. We know that $\sin^2(\theta) = \frac{1-\cos(2\theta)}{2}$. So, we can write: $$\sin^2\left(\theta + \frac{\pi}{6}\right) = \frac{1-\cos\left(2\left(\theta + \frac{\pi}{6}\right)\right)}{2}$$. Now, we can simplify the argument of the cosine function: $$2\left(\theta + \frac{\pi}{6}\right) = 2\theta + \frac{\pi}{3}$$. Therefore, the integral becomes: $$\int \sin^2\left(\theta+\frac{\pi}{6}\right) d\theta = \int \frac{1-\cos(2\theta + \frac{\pi}{3})}{2} d\theta$$.

Distribute the integral

Now, we distribute the integral into two separate integrals: $$\int \frac{1-\cos(2\theta + \frac{\pi}{3})}{2} d\theta = \frac{1}{2}\int 1 d\theta - \frac{1}{2}\int \cos(2\theta + \frac{\pi}{3}) d\theta$$.

Integrate the first integral

Integrating the first integral with respect to $\theta$ is straightforward: $$\frac{1}{2}\int 1 d\theta = \frac{1}{2}\theta + C_1$$.

Integrate the second integral using substitution

To integrate the second integral, we will use the substitution method. Let $u = 2\theta + \frac{\pi}{3}$. Then, $du = 2 d\theta$ and $d\theta = \frac{1}{2} du$. So, we have: $$-\frac{1}{2}\int \cos(2\theta + \frac{\pi}{3}) d\theta = -\frac{1}{2}\int \cos(u) \left(\frac{1}{2} du\right) = -\frac{1}{4}\int \cos(u) du$$. Now, integrate with respect to $u$: $$-\frac{1}{4}\int \cos(u) du = -\frac{1}{4} \sin(u) + C_2$$. Finally, replace $u$ with the original expression: $$-\frac{1}{4} \sin(u) + C_2 = -\frac{1}{4} \sin(2\theta +\frac{\pi}{3}) + C_2$$.

Combine the results

Now, we combine the results of both integrals to obtain the final result. The integral of $\sin^2(\theta+\frac{\pi}{6})$ with respect to $\theta$ is: $$\frac{1}{2}\theta - \frac{1}{4} \sin(2\theta +\frac{\pi}{3}) + C$$, where $C = C_1 + C_2$ is the constant of integration.

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Integrals with \(\sin ^{2} x\) and \(\cos ^{2} x\) Evaluate the following integrals. $$\int \sin ^{2}\left(\theta+\frac{\pi}{6}\right) d \theta$$

Short Answer

Step by step solution

Apply the double-angle formula

Distribute the integral

Integrate the first integral

Integrate the second integral using substitution

Combine the results

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