Evaluate the Integral

Problem

(∫_^)((9−2017*x2⋅16+6√(5,x)+12*e(4*x)−5/x)*d(x))

Solution

1. Simplify the constant coefficients in the second term by multiplying 2017 and 16

2017⋅16=32272

2. Rewrite the radical term using a fractional exponent and the fraction term using a standard form for integration.

√(5,x)=x(1/5)

3. Apply the sum rule for integration to break the expression into individual integrals.

(∫_^)(9*d(x))−(∫_^)(32272*x2*d(x))+(∫_^)(6*x(1/5)*d(x))+(∫_^)(12*e(4*x)*d(x))−(∫_^)(5/x*d(x))

4. Integrate each term using the power rule (∫_^)(xn*d(x))=(x(n+1))/(n+1) the exponential rule (∫_^)(e(a*x)*d(x))=(e(a*x))/a and the logarithmic rule (∫_^)(1/x*d(x))=ln(x)

(∫_^)(9*d(x))=9*x

(∫_^)(32272*x2*d(x))=(32272*x3)/3

(∫_^)(6*x(1/5)*d(x))=6⋅(x(6/5))/(6/5)=6⋅5/6*x(6/5)=5*x(6/5)

(∫_^)(12*e(4*x)*d(x))=(12*e(4*x))/4=3*e(4*x)

(∫_^)(5/x*d(x))=5*ln(x)

5. Combine all the results and add the constant of integration C

9*x−(32272*x3)/3+5*x(6/5)+3*e(4*x)−5*ln(x)+C

Final Answer

(∫_^)((9−2017*x2⋅16+6√(5,x)+12*e(4*x)−5/x)*d(x))=9*x−(32272*x3)/3+5*x(6/5)+3*e(4*x)−5*ln(x)+C

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