Evaluate the Integral

Problem

(∫_^)(2*sin(x)+3*cos(x)*d(x))

Solution

1. Use the sum rule for integration to split the integral into two separate terms.

(∫_^)(2*sin(x)+3*cos(x)*d(x))=(∫_^)(2*sin(x)*d(x))+(∫_^)(3*cos(x)*d(x))

2. Factor out the constants from each integral.

(∫_^)(2*sin(x)*d(x))+(∫_^)(3*cos(x)*d(x))=2*(∫_^)(sin(x)*d(x))+3*(∫_^)(cos(x)*d(x))

3. Apply the basic integration rules for trigonometric functions, where (∫_^)(sin(x)*d(x))=−cos(x) and (∫_^)(cos(x)*d(x))=sin(x)

2*(−cos(x))+3*(sin(x))+C

4. Simplify the expression and include the constant of integration C

−2*cos(x)+3*sin(x)+C

Final Answer

(∫_^)(2*sin(x)+3*cos(x)*d(x))=−2*cos(x)+3*sin(x)+C

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