Evaluate the Integral

Problem

(∫_^)(9*θ*cos(θ)*d(θ))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic term 9*θ and a trigonometric term cos(θ) use integration by parts.

2. Choose the parts for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=9*θ and d(v)=cos(θ)*d(θ)

3. Differentiate u and integrate d(v) to find d(u) and v

d(u)=9*d(θ)

v=sin(θ)

4. Apply the integration by parts formula.

(∫_^)(9*θ*cos(θ)*d(θ))=(9*θ)*(sin(θ))−(∫_^)(sin(θ)*(9*d(θ)))

5. Simplify the expression and evaluate the remaining integral.

(∫_^)(9*θ*cos(θ)*d(θ))=9*θ*sin(θ)−9*(∫_^)(sin(θ)*d(θ))

6. Integrate sin(θ) noting that (∫_^)(sin(θ)*d(θ))=−cos(θ)

(∫_^)(9*θ*cos(θ)*d(θ))=9*θ*sin(θ)−9*(−cos(θ))+C

7. Simplify the signs to reach the final result.

(∫_^)(9*θ*cos(θ)*d(θ))=9*θ*sin(θ)+9*cos(θ)+C

Final Answer

(∫_^)(9*θ*cos(θ)*d(θ))=9*θ*sin(θ)+9*cos(θ)+C

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