/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Find the indefinite integral. ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 19

Find the indefinite integral. $$ \int \csc 2 x d x $$

Short Answer

Expert verified
The indefinite integral of \(\csc(2x)\) is \(-1/2 \ln | \csc(2x) + \cot(2x) | + C\).

Step by step solution

01

Recognize the integral

Here, we need to recognize the integral of \(\csc(2x)\). It's also important to note that the variable 'x' is being multiplied by 2 inside the cosecant function. Therefore, we will need to adjust for this factor later on.
02

Apply the formula for integration of csc(u) functions

The integral of \(\csc(u)\) is \(-\ln | \csc(u) + \cot(u) | + C\), where 'u' is the function inside the cosecant function. Thus, in our case, the integral becomes \(-\ln | \csc(2x) + \cot(2x) |\).
03

Adjust for the factor of 2

Because the integral formula applied in Step 2 assumes that 'u' is simply 'x' and not '2x', we need to adjust for this. The rule for adjusting is simple: if you have a function 2x instead of just x, you divide the whole expression by the multiple of x, which in this case is '2'. So, the final integral becomes \(-1/2 \ln | \csc(2x) + \cot(2x) | + C\).

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