Find the Antiderivative h(theta)=2sin(theta)-sec(theta)^2
The problem is asking for the calculation of the antiderivative (or indefinite integral) of the function h(θ) = 2sin(θ) - sec(θ)^2 with respect to the variable θ. The symbols "sin" and "sec" denote the sine and secant trigonometric functions, respectively. The secant function, sec(θ), is the reciprocal of the cosine function, meaning sec(θ) = 1/cos(θ). The antiderivative of a function is another function whose derivative gives the original function. The process of finding the antiderivative is the reverse of taking a derivative and is a fundamental operation in calculus often used to calculate areas or solve differential equations. The antiderivative is not unique; rather it is defined up to an arbitrary constant of integration, which is usually represented by the letter C.
$h \left(\right. \theta\left.\right) = 2 sin \left(\right. \theta\left.\right) - \left(sec\right)^{2} \left(\right. \theta\left.\right)$
Answer
Solution:
Step 1:
Identify the antiderivative $H(\theta)$ by integrating the function $h(\theta)$. $$H(\theta) = \int h(\theta) d\theta$$
Step 2:
Write down the integral that needs to be solved.
$$H(\theta) = \int (2 \sin(\theta) - \sec^2(\theta)) d\theta$$
Step 3:
Decompose the integral into two separate integrals.
$$\int 2 \sin(\theta) d\theta - \int \sec^2(\theta) d\theta$$
Step 4:
Extract the constant $2$ from the first integral.
$$2 \int \sin(\theta) d\theta - \int \sec^2(\theta) d\theta$$
Step 5:
Integrate $\sin(\theta)$ to get $- \cos(\theta)$.
$$2(-\cos(\theta) + C_1) - \int \sec^2(\theta) d\theta$$
Step 6:
Factor out the negative sign from the second integral.
$$2(-\cos(\theta) + C_1) - \int \sec^2(\theta) d\theta$$
Step 7:
Recognize that the integral of $\sec^2(\theta)$ is $\tan(\theta)$.
$$2(-\cos(\theta) + C_1) - (\tan(\theta) + C_2)$$
Step 8:
Combine terms and constants.
$$-2\cos(\theta) - \tan(\theta) + C$$
Step 9:
Present the final antiderivative of $h(\theta)$.
$$H(\theta) = -2\cos(\theta) - \tan(\theta) + C$$
Knowledge Notes:
The process of finding the antiderivative, also known as the indefinite integral, involves reversing the differentiation process. Here are the relevant knowledge points:
Indefinite Integral: The antiderivative or indefinite integral of a function $f(x)$ is denoted by $\int f(x) dx$ and represents the family of all functions whose derivative is $f(x)$.
Linearity of Integration: The integral of a sum or difference of functions is the sum or difference of their integrals. This property allows us to split the integral of $2\sin(\theta) - \sec^2(\theta)$ into two separate integrals.
Constant Multiple Rule: A constant multiplier can be factored out of an integral. This is used to pull the $2$ out of the integral of $2\sin(\theta)$.
Basic Trigonometric Integrals: The antiderivatives of basic trigonometric functions are known, such as:
- $\int \sin(\theta) d\theta = -\cos(\theta) + C$
- $\int \sec^2(\theta) d\theta = \tan(\theta) + C$ where $C$ represents the constant of integration.
Combining Constants: When combining integrals, the constants of integration can be combined into a single constant, since they represent an arbitrary value.
Simplification: After integrating, it's important to simplify the expression to obtain the final result. This may involve combining like terms and constants.
Understanding these principles is essential for solving integration problems and finding antiderivatives.
