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

Problem

(∫_^)(tan(x)*d(x))

Solution

1. Rewrite the tangent function in terms of sine and cosine using the trigonometric identity tan(x)=sin(x)/cos(x)

(∫_^)(sin(x)/cos(x)*d(x))

2. Identify a substitution by letting u=cos(x)

3. Differentiate u to find d(u) which gives d(u)/d(x)=−sin(x) or d(u)=−sin(x)*d(x)

−d(u)=sin(x)*d(x)

4. Substitute u and d(u) into the integral.

(∫_^)(−1/u*d(u))

5. Integrate the expression using the rule (∫_^)(1/u*d(u))=ln(u)+C

−ln(u)+C

6. Back-substitute u=cos(x) to return to the original variable.

−ln(cos(x))+C

7. Simplify the expression using the logarithm power rule −ln(a)=ln(a(−1)) noting that 1/cos(x)=sec(x)

ln(sec(x))+C

Final Answer

(∫_^)(tan(x)*d(x))=ln(sec(x))+C

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