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Math  /  Calculus

QuestionEvaluate the integral ∫5csc⁡(x)dx\int 5 \csc (x) d x

Studdy Solution

STEP 1

What is this asking? We need to find the *indefinite integral* of 5csc⁡(x)5 \csc(x) with respect to xx.
Basically, we're looking for a function whose derivative is 5csc⁡(x)5 \csc(x). Watch out! Remember that there are a few different ways to write the solution to this integral, and they might look different even though they're equivalent!
Don't panic if your answer doesn't immediately match what you see in a textbook or online.

STEP 2

1. Rewrite the integral
2. Multiply by a clever form of 1
3. Use *u*-substitution
4. Integrate
5. Substitute back

STEP 3

Let's **rewrite** our integral using the definition of csc⁡(x)\csc(x): ∫5csc⁡(x) dx=∫5sin⁡(x) dx \int 5 \csc(x) \, dx = \int \frac{5}{\sin(x)} \, dx This makes it a bit easier to work with!

STEP 4

Now for a little magic!
We're going to multiply the integrand by a **clever form of 1**: csc⁡(x)+cot⁡(x)csc⁡(x)+cot⁡(x)\frac{\csc(x) + \cot(x)}{\csc(x) + \cot(x)}.
It might seem strange, but trust me, it'll make things work out beautifully! ∫5sin⁡(x) dx=∫5sin⁡(x)⋅csc⁡(x)+cot⁡(x)csc⁡(x)+cot⁡(x) dx=∫5(csc⁡(x)+cot⁡(x))sin⁡(x)(csc⁡(x)+cot⁡(x)) dx \int \frac{5}{\sin(x)} \, dx = \int \frac{5}{\sin(x)} \cdot \frac{\csc(x) + \cot(x)}{\csc(x) + \cot(x)} \, dx = \int \frac{5(\csc(x) + \cot(x))}{\sin(x)(\csc(x) + \cot(x))} \, dx

STEP 5

Let's **simplify** the denominator: sin⁡(x)(csc⁡(x)+cot⁡(x))=sin⁡(x)csc⁡(x)+sin⁡(x)cot⁡(x)=sin⁡(x)⋅1sin⁡(x)+sin⁡(x)⋅cos⁡(x)sin⁡(x)=1+cos⁡(x) \sin(x)(\csc(x) + \cot(x)) = \sin(x) \csc(x) + \sin(x) \cot(x) = \sin(x) \cdot \frac{1}{\sin(x)} + \sin(x) \cdot \frac{\cos(x)}{\sin(x)} = 1 + \cos(x) So our integral becomes: ∫5(csc⁡(x)+cot⁡(x))1+cos⁡(x) dx \int \frac{5(\csc(x) + \cot(x))}{1 + \cos(x)} \, dx

STEP 6

Time for *u*-substitution!
Let u=1+cos⁡(x)u = 1 + \cos(x).
Then du=−sin⁡(x) dxdu = -\sin(x) \, dx.
We also know that csc⁡(x)+cot⁡(x)=1+cos⁡(x)sin⁡(x)=u−sin⁡(x) dx=−udu\csc(x) + \cot(x) = \frac{1 + \cos(x)}{\sin(x)} = \frac{u}{-\sin(x) \, dx} = -\frac{u}{du}.

STEP 7

Substituting these into our integral, we get: ∫5(csc⁡(x)+cot⁡(x))1+cos⁡(x) dx=∫5u⋅u−du=∫−5du du=−5∫1u du \int \frac{5(\csc(x) + \cot(x))}{1 + \cos(x)} \, dx = \int \frac{5}{u} \cdot \frac{u}{-du} = \int -\frac{5}{du} \, du = -5 \int \frac{1}{u} \, du

STEP 8

Now, we can **integrate**!
The integral of 1u\frac{1}{u} is ln⁡∣u∣\ln|u|, so we have: −5∫1u du=−5ln⁡∣u∣+C -5 \int \frac{1}{u} \, du = -5 \ln|u| + C where CC is the constant of integration.

STEP 9

Finally, let's **substitute** back u=1+cos⁡(x)u = 1 + \cos(x): −5ln⁡∣u∣+C=−5ln⁡∣1+cos⁡(x)∣+C -5 \ln|u| + C = -5 \ln|1 + \cos(x)| + C

STEP 10

Our final answer is −5ln⁡∣1+cos⁡(x)∣+C-5 \ln|1 + \cos(x)| + C.

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