Find the indefinite integral. (Use C for the constant of integration.) csc(x) cot(x) V 16 csc2 X
The Correct Answer and Explanation is:
To solve the indefinite integral of the expression:∫csc(x)cot(x)⋅16csc2(x) dx\int \csc(x) \cot(x) \cdot \sqrt{16 \csc^2(x)} \, dx∫csc(x)cot(x)⋅16csc2(x)dx
Step 1: Simplify the expression
We start by simplifying the square root term:16csc2(x)=4csc(x)\sqrt{16 \csc^2(x)} = 4 \csc(x)16csc2(x)=4csc(x)
Thus, the integral becomes:∫csc(x)cot(x)⋅4csc(x) dx\int \csc(x) \cot(x) \cdot 4 \csc(x) \, dx∫csc(x)cot(x)⋅4csc(x)dx
This simplifies further to:∫4csc2(x)cot(x) dx\int 4 \csc^2(x) \cot(x) \, dx∫4csc2(x)cot(x)dx
Step 2: Make a substitution
Notice that the integral contains both csc2(x)\csc^2(x)csc2(x) and cot(x)\cot(x)cot(x), which suggests using substitution. We will use the substitution:u=cot(x)u = \cot(x)u=cot(x)
Thus, the derivative of cot(x)\cot(x)cot(x) with respect to xxx is:dudx=−csc2(x)\frac{du}{dx} = -\csc^2(x)dxdu=−csc2(x)
or equivalently:du=−csc2(x) dxdu = -\csc^2(x) \, dxdu=−csc2(x)dx
Step 3: Substitute and integrate
Substitute u=cot(x)u = \cot(x)u=cot(x) and du=−csc2(x) dxdu = -\csc^2(x) \, dxdu=−csc2(x)dx into the integral:∫4csc2(x)cot(x) dx=−4∫u du\int 4 \csc^2(x) \cot(x) \, dx = -4 \int u \, du∫4csc2(x)cot(x)dx=−4∫udu
Now, we can integrate with respect to uuu:−4∫u du=−4⋅u22=−2u2-4 \int u \, du = -4 \cdot \frac{u^2}{2} = -2u^2−4∫udu=−4⋅2u2=−2u2
Step 4: Substitute back
Now, we substitute u=cot(x)u = \cot(x)u=cot(x) back into the expression:−2u2=−2cot2(x)-2u^2 = -2 \cot^2(x)−2u2=−2cot2(x)
Step 5: Add the constant of integration
Finally, since we are evaluating an indefinite integral, we add the constant of integration CCC:∫csc(x)cot(x)⋅16csc2(x) dx=−2cot2(x)+C\int \csc(x) \cot(x) \cdot \sqrt{16 \csc^2(x)} \, dx = -2 \cot^2(x) + C∫csc(x)cot(x)⋅16csc2(x)dx=−2cot2(x)+C
Final Answer:
−2cot2(x)+C\boxed{-2 \cot^2(x) + C}−2cot2(x)+C
Explanation:
In this solution, we used substitution to simplify the integral. By recognizing the relationship between the trigonometric functions and using the derivative of cot(x)\cot(x)cot(x), we were able to transform the integral into a more straightforward form, which was then solved using basic integration rules. The constant of integration CCC accounts for the fact that the integral represents a family of functions, all differing by a constant.
