Evaluate the Integral integral of sec(x)^3 with respect to x

Problem

(∫_^)(sec3(x)*d(x))

Solution

1. Apply integration by parts by setting u=sec(x) and d(v)=sec2(x)*d(x)

2. Differentiate and integrate to find d(u)=sec(x)*tan(x)*d(x) and v=tan(x)

3. Substitute into the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

(∫_^)(sec3(x)*d(x))=sec(x)*tan(x)−(∫_^)(sec(x)*tan2(x)*d(x))

4. Use the trigonometric identity tan2(x)=sec2(x)−1 to rewrite the integral.

(∫_^)(sec3(x)*d(x))=sec(x)*tan(x)−(∫_^)(sec(x)*(sec2(x)−1)*d(x))

5. Distribute and split the integral on the right side.

(∫_^)(sec3(x)*d(x))=sec(x)*tan(x)−(∫_^)(sec3(x)*d(x))+(∫_^)(sec(x)*d(x))

6. Add the integral (∫_^)(sec3(x)*d(x)) to both sides of the equation.

2*(∫_^)(sec3(x)*d(x))=sec(x)*tan(x)+(∫_^)(sec(x)*d(x))

7. Evaluate the integral of sec(x) which is ln(sec(x)+tan(x))

2*(∫_^)(sec3(x)*d(x))=sec(x)*tan(x)+ln(sec(x)+tan(x))

8. Divide by 2 and add the constant of integration C

(∫_^)(sec3(x)*d(x))=1/2*sec(x)*tan(x)+1/2*ln(sec(x)+tan(x))+C

Final Answer

(∫_^)(sec3(x)*d(x))=1/2*(sec(x)*tan(x)+ln(sec(x)+tan(x)))+C

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