Evaluate the Integral integral of tan(4x) with respect to x

Problem

(∫_^)(tan(4*x)*d(x))

Solution

1. Use the identity for the tangent function to rewrite the integral in terms of sine and cosine.

(∫_^)(tan(4*x)*d(x))=(∫_^)(sin(4*x)/cos(4*x)*d(x))

2. Identify a substitution by letting u be the denominator.

u=cos(4*x)

3. Find the differential d(u) by differentiating u with respect to x

d(u)/d(x)=−4*sin(4*x)

d(u)=−4*sin(4*x)*d(x)

4. Adjust the differential to match the numerator in the integral.

−1/4*d(u)=sin(4*x)*d(x)

5. Substitute the variables u and d(u) into the integral.

(∫_^)(sin(4*x)/cos(4*x)*d(x))=(∫_^)(1/u*(−1/4)*d(u))

6. Factor out the constant and integrate using the natural log rule.

−1/4*(∫_^)(1/u*d(u))=−1/4*ln(u)+C

7. Back-substitute the original expression for u to get the final result.

−1/4*ln(cos(4*x))+C

Final Answer

(∫_^)(tan(4*x)*d(x))=−1/4*ln(cos(4*x))+C

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