Evaluate the Integral

Problem

(∫_^)(e(2*x)*(1+e(2*x))3*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, noting that the derivative of the inner function 1+e(2*x) is a multiple of the outer factor e(2*x)

2. Substitute u=1+e(2*x)

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

d(u)/d(x)=2*e(2*x)

d(u)=2*e(2*x)*d(x)

1/2*d(u)=e(2*x)*d(x)

4. Rewrite the integral in terms of u

(∫_^)(u3⋅1/2*d(u))

1/2*(∫_^)(u3*d(u))

5. Integrate with respect to u using the power rule.

1/2⋅(u4)/4+C

(u4)/8+C

6. Back-substitute the original expression for u to get the final result in terms of x

((1+e(2*x))4)/8+C

Final Answer

(∫_^)(e(2*x)*(1+e(2*x))3*d(x))=((1+e(2*x))4)/8+C

Want more problems? Check here!