Evaluate the Integral

Problem

(∫_^)(sec2(x)/√(,tan(x))*d(x))

Solution

1. Identify the substitution by noticing that the derivative of tan(x) is sec2(x) which appears in the numerator.

2. Substitute u=tan(x) which implies d(u)=sec2(x)*d(x)

3. Rewrite the integral in terms of u

(∫_^)(1/√(,u)*d(u))

4. Express the integrand as a power of u

(∫_^)(u(−1/2)*d(u))

5. Apply the power rule for integration, (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u(1/2))/(1/2)+C

6. Simplify the expression:

2√(,u)+C

7. Back-substitute u=tan(x) to return to the original variable.

Final Answer

(∫_^)(sec2(x)/√(,tan(x))*d(x))=2√(,tan(x))+C

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