Evaluate the Integral

Problem

(∫_^)(1/√(,x^3)+5*e^(5*x)*d(x))

Solution

1. Rewrite the first term using exponent rules to make it easier to integrate, changing 1/√(,x^3) 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 (∫_^)(x^n*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/√(,x^3)+5*e^(5*x)*d(x))=−2/√(,x)+e^(5*x)+C

Want more problems? Check here!