Evaluate the Integral integral of 7xe^(5x^2) with respect to x

Problem

(∫_^)(7*x*e(5*x2)*d(x))

Solution

1. Identify the substitution method as appropriate because the exponent 5*x2 has a derivative that is a multiple of the x term outside the exponential.

2. Substitute u=5*x2 to simplify the integrand.

3. Differentiate u with respect to x to find d(u)=10*x*d(x)

4. Rearrange the differential to solve for x*d(x) resulting in x*d(x)=1/10*d(u)

5. Rewrite the integral in terms of u by pulling out the constant 7 and substituting the expressions for u and d(x)

(∫_^)(7*x*e(5*x2)*d(x))=7*(∫_^)(eu⋅1/10*d(u))

6. Simplify the integral by moving the constant factor 1/10 outside.

7/10*(∫_^)(eu*d(u))

7. Integrate the expression with respect to u remembering that the integral of eu is eu

7/10*eu+C

8. Back-substitute the original expression 5*x2 for u to get the final result in terms of x

Final Answer

(∫_^)(7*x*e(5*x2)*d(x))=(7*e(5*x2))/10+C

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