$\int \frac{x+5}{(x+6)^2} e^x \, dx$ is equal to:

To solve the integral ∫ ( x + 6 ) 2 x + 5 ​ e x d x , we can apply integration techniques such as integration by parts or simplifying the expression.
One suitable method for this integral is integration by parts, which is useful when dealing with products of functions. The formula for integration by parts is:
∫ u d v = uv − ∫ v d u
We need to identify parts of the integrand as u and d v . Let's choose:

u = ( x + 6 ) 2 x + 5 ​ , which means d u is the derivative of u .

d v = e x d x , which means v = e x .


Now, let's calculate the derivative d u . Using the quotient rule:
d x d ​ ( ( x + 6 ) 2 x + 5 ​ ) = (( x + 6 ) 2 ) 2 ( x + 6 ) 2 ( 1 ) − ( x + 5 ) ( 2 ) ( x + 6 ) ​
Simplifying the numerator:
= ( x + 6 ) 2 − 2 ( x + 5 ) ( x + 6 )
After finding the derivative and simplifying, use the parts in the integration by parts formula:
∫ ( x + 6 ) 2 x + 5 ​ e x d x = ( x + 6 ) 2 x + 5 ​ e x − ∫ e x ( result from d u ) d x
This process involves some complex algebra, so make sure you simplify correctly and follow through with integrating by parts. You may need to apply integration by parts a second time or use substitution if the integral requires further simplification.
This typical college-level integral requires practice and familiarity with techniques like integration by parts and substitutions. If further assistance or step-by-step simplification is needed, feel free to ask!

Answered by danjohnbrain | 2025-07-06