Evaluate the Integral

Problem

(∫_^)(1/√(,x3)+5*e(5*x)*d(x))

Solution

1. Rewrite the first term using exponent rules to make it easier to integrate, changing 1/√(,x3) to x(−3/2)

(∫_^)(x(−3/2)+5*e(5*x)*d(x))

2. Apply the sum rule for integration to evaluate each term separately.

(∫_^)(x(−3/2)*d(x))+(∫_^)(5*e(5*x)*d(x))

3. Use the power rule (∫_^)(xn*d(x))=(x(n+1))/(n+1) for the first term.

(x(−1/2))/(−1/2)=−2*x(−1/2)

4. Use the exponential rule (∫_^)(e(a*x)*d(x))=1/a*e(a*x) for the second term.

5⋅1/5*e(5*x)=e(5*x)

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

−2*x(−1/2)+e(5*x)+C

6. Simplify the expression by rewriting the negative exponent as a radical in the denominator.

−2/√(,x)+e(5*x)+C

Final Answer

(∫_^)(1/√(,x3)+5*e(5*x)*d(x))=−2/√(,x)+e(5*x)+C

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