Integrals of Trig Functions: Videos & Practice Problems

Integrating trigonometric functions involves reversing their derivatives. For example, the integral of $\cos \left(x\right)$ is $\sin \left(x\right)$ plus a constant, while the integral of $\sin \left(x\right)$ is $-\cos \left(x\right)$ plus a constant. Using the sum and difference rule, constants can be factored out, simplifying integration of expressions like $7sec\left(x\right)tan\left(x\right)$ and $-\mathrm{cosec}\left(x\right)$.

Integrals Resulting in Basic Trig Functions

In the study of integrals, particularly with trigonometric functions, it's essential to understand that integrating these functions is essentially the reverse process of differentiation. For instance, the integral of the cosine function can be derived from the fact that the derivative of sine is cosine. Therefore, the integral of cos(x) is sin(x) + C, where C represents the constant of integration. Similarly, since the derivative of cosine is negative sine, the integral of sin(x) results in -cos(x) + C.

When dealing with more complex expressions, such as the integral of 3sin(x) + 2cos(x), we can apply the sum and difference rule. This allows us to separate the integral into two parts: ∫3sin(x)dx + ∫2cos(x)dx. By applying the constant multiple rule, we can factor out the constants, leading to 3∫sin(x)dx + 2∫cos(x)dx. The integrals of sine and cosine yield -3cos(x) + 2sin(x) + C as the final result.

For integrals involving other trigonometric functions, such as 7sec(x)tan(x) - cosec^2(x), we again utilize the sum and difference rule to split the integral. This gives us 7∫sec(x)tan(x)dx - ∫cosec^2(x)dx. Recognizing that the derivative of sec(x) is sec(x)tan(x), we find that the integral of sec(x)tan(x) is sec(x). For cosec^2(x), knowing that the derivative of cot(x) is -cosec^2(x) allows us to conclude that the integral of cosec^2(x) is -cot(x). Thus, the final result for this integral is 7sec(x) + cot(x) + C.

To effectively tackle integrals involving trigonometric functions, it is crucial to memorize the derivatives of these functions, as they provide a straightforward method for finding their integrals. For example, the derivative of tan(x) is sec^2(x), leading to the integral of sec^2(x) being tan(x) + C. Similarly, the derivative of cosec(x) is -cosec(x)cot(x), which implies that the integral of cosec(x)cot(x) is -cosec(x) + C.

By applying these principles and recognizing the relationships between derivatives and integrals, students can confidently approach problems involving trigonometric integrals.

Find $g\left(\theta\right)$ by evaluating the following indefinite integral.
$g\left(\theta\right)=\int^{}\left(5\sec^2\theta-2\csc^2\theta\right)d\theta$

• A. $g\left(\theta\right)=5\tan\theta+2\cot\theta+C$
• B. $g\left(\theta\right)=5\tan\theta-2\cot\theta+C$
• C. $g\left(\theta\right)=5\cot\theta+2\tan\theta+C$
• D. $g\left(\theta\right)=5\cot\theta-2\tan\theta+C$

Answer: (A)

Find $g\left(x\right)$ by evaluating the following indefinite integral.
$g\left(x\right)=\int\left(\sin^2x-100\csc x\cot x+\cos^2x\right)dx$

• A. $g\left(x\right)=100\csc x+x+C$
• B. $g\left(x\right)=-100\csc x+x+C$
• C. $g\left(x\right)=100\csc x+C$
• D. $g\left(x\right)=-100\csc x+C$

Answer: (A)

Integrals Resulting in Basic Trig Functions Example 1

In this problem, we are tasked with verifying an indefinite integral by differentiation. The process involves taking the derivative of the proposed solution to the integral and checking if it matches the original integrand. This method essentially reverses the integration process, as finding an indefinite integral is akin to determining an antiderivative.

To begin, we differentiate the expression given as the solution, which is \(2 \sin(\sqrt{x}) + C\), where \(C\) is the constant of integration. According to the rules of differentiation, we can apply the derivative to each term separately. The derivative of a constant \(C\) is zero, so we focus on differentiating \(2 \sin(\sqrt{x})\).

Using the chain rule, we first recognize that the derivative of \(\sin(u)\) is \(\cos(u)\), where \(u = \sqrt{x}\). Thus, we have:

\[\frac{d}{dx}[2 \sin(\sqrt{x})] = 2 \cos(\sqrt{x}) \cdot \frac{d}{dx}[\sqrt{x}]\]

Next, we need to find the derivative of \(\sqrt{x}\). This can be expressed as \(x^{1/2}\). Applying the power rule, we get:

\[\frac{d}{dx}[x^{1/2}] = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}\]

Substituting this back into our derivative calculation gives us:

\[\frac{d}{dx}[2 \sin(\sqrt{x})] = 2 \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} = \frac{\cos(\sqrt{x})}{\sqrt{x}}\]

Now, we compare this result with the original integrand. The expression \(\frac{\cos(\sqrt{x})}{\sqrt{x}}\) matches the integrand, confirming that our differentiation process is correct. Therefore, we have successfully verified that the indefinite integral is accurate, as the derivative of the proposed solution yields the original function we started with.

This verification process illustrates the fundamental relationship between integration and differentiation, reinforcing the concept that integration is the reverse operation of differentiation.

Here’s what students ask on this topic:

What is the integral of cos(x)?

The integral of cos(x) is sin(x) plus a constant of integration. In mathematical terms, this is expressed as:

$\int \cos \left(x\right)dx=\sin \left(x\right)+C$

Here, C represents the constant of integration, which is added to account for any constant value that could have been present before differentiation.

How do you integrate sin(x)?

To integrate sin(x), you need to recognize that the integral of sin(x) is -cos(x) plus a constant of integration. This can be written as:

$\int \sin \left(x\right)dx=-\cos \left(x\right)+C$

Again, C is the constant of integration, which accounts for any constant that could have been present before differentiation.

What is the integral of sec(x)tan(x)?

The integral of sec(x)tan(x) is sec(x) plus a constant of integration. This is because the derivative of sec(x) is sec(x)tan(x). Therefore, the integral is:

$\int \sec \left(x\right)\tan \left(x\right)dx=\sec \left(x\right)+C$

Here, C is the constant of integration.

How do you integrate cosec^2(x)?

The integral of cosec^2(x) is -cot(x) plus a constant of integration. This is because the derivative of cot(x) is -cosec^2(x). Therefore, the integral is:

$\int \mathrm{cosec}^2\left(x\right)dx=-\cot \left(x\right)+C$

Here, C is the constant of integration.

What is the integral of 7sec(x)tan(x) - cosec^2(x)?

To integrate 7sec(x)tan(x) - cosec^2(x), we use the sum and difference rule and the constant multiple rule. The integral is:

$\int 7\sec \left(x\right)\tan \left(x\right)-\mathrm{cosec}^2\left(x\right)dx=7\sec \left(x\right)+-\cot \left(x\right)+C$

Here, C is the constant of integration.