Find the Integral cos(5x)

Problem

(∫_^)(cos(5*x)*d(x))

Solution

1. Identify the integral as a basic trigonometric form (∫_^)(cos(a*x)*d(x)) where a=5

2. Apply the substitution u=5*x which implies d(u)=5*d(x) or d(x)=1/5*d(u)

3. Substitute these into the integral to get (∫_^)(cos(u)⋅1/5*d(u))

4. Integrate the cosine function, recalling that (∫_^)(cos(u)*d(u))=sin(u)+C

5. Back-substitute u=5*x into the result and include the constant of integration C

Final Answer

(∫_^)(cos(5*x)*d(x))=1/5*sin(5*x)+C

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