Evaluate. (Assume $x>0$.) Check by differentiating.
$\int(x+9) \ln x d x$
$\int(x+9) \ln x d x=$
Answer (1)
Apply integration by parts with u = ln x and d v = ( x + 9 ) d x .
Find d u = x 1 d x and v = 2 x 2 + 9 x .
Use the integration by parts formula: ∫ u d v = uv − ∫ v d u to get ∫ ( x + 9 ) ln x d x = ( 2 x 2 + 9 x ) ln x − ∫ ( 2 x + 9 ) d x .
Evaluate the remaining integral and simplify to obtain ( 2 x 2 + 9 x ) ln x − 4 x 2 − 9 x + C .
Explanation
Problem Analysis We are asked to evaluate the indefinite integral ∫ ( x + 9 ) ln x d x and check the result by differentiation.
Applying Integration by Parts We will use integration by parts. The formula for integration by parts is ∫ u d v = uv − ∫ v d u . Let u = ln x and d v = ( x + 9 ) d x . Then, we find d u and v .
Finding du and v Differentiating u = ln x with respect to x , we get d u = x 1 d x . Integrating d v = ( x + 9 ) d x with respect to x , we get v = ∫ ( x + 9 ) d x = 2 x 2 + 9 x .
Substituting into the Formula Now, we substitute u , v , d u , and d v into the integration by parts formula:
Applying the Formula ∫ ( x + 9 ) ln x d x = ( 2 x 2 + 9 x ) ln x − ∫ ( 2 x 2 + 9 x ) x 1 d x
Simplifying the Integral Simplify the integral:
Simplifying the Integral ∫ ( 2 x 2 + 9 x ) x 1 d x = ∫ ( 2 x + 9 ) d x
Evaluating the Simplified Integral Evaluate the simplified integral:
Evaluating the Simplified Integral ∫ ( 2 x + 9 ) d x = 4 x 2 + 9 x + C where C is the constant of integration.
Substituting Back Substitute the result back into the integration by parts formula:
Substituting Back ∫ ( x + 9 ) ln x d x = ( 2 x 2 + 9 x ) ln x − ( 4 x 2 + 9 x ) + C
Simplifying the Expression Simplify the expression:
Simplifying the Expression ∫ ( x + 9 ) ln x d x = ( 2 x 2 + 9 x ) ln x − 4 x 2 − 9 x + C
Checking by Differentiation Check by differentiating the result:
Differentiating the Result d x d [ ( 2 x 2 + 9 x ) ln x − 4 x 2 − 9 x + C ] = ( x + 9 ) ln x + ( 2 x 2 + 9 x ) x 1 − 2 x − 9 = ( x + 9 ) ln x + 2 x + 9 − 2 x − 9 = ( x + 9 ) ln x
Verification The derivative of the result matches the original integrand, so the integration is correct.
Final Answer Therefore, the indefinite integral is:
Final Answer ∫ ( x + 9 ) ln x d x = ( 2 x 2 + 9 x ) ln x − 4 x 2 − 9 x + C
Examples
Imagine you're calculating the total cost of materials for a project where the price of a material increases logarithmically with the amount purchased. Integrating a function that includes a logarithmic term, similar to this problem, helps you determine the overall expenditure, ensuring accurate budget planning and resource allocation.
