Evaluate the Integral integral of arcsec(x) with respect to x

Problem

(∫_^)(arcsec(x)*d(x))

Solution

1. Identify the method of integration by parts, where (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign the variables for integration by parts by letting u=arcsec(x) and d(v)=d(x)

3. Differentiate u to find d(u)=1/(|x|√(,x2−1))*d(x) and integrate d(v) to find v=x

4. Substitute these into the integration by parts formula to get x*arcsec(x)−(∫_^)(x/(|x|√(,x2−1))*d(x))

5. Simplify the integral for x>1 where x/|x|=1 resulting in (∫_^)(1/√(,x2−1)*d(x))

6. Apply the standard integral formula for (∫_^)(1/√(,x2−1)*d(x)) which is ln(x+√(,x2−1))

7. Combine the results and add the constant of integration C

Final Answer

(∫_^)(arcsec(x)*d(x))=x*arcsec(x)−ln(x+√(,x2−1))+C

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