Consider the indefinite integral I= x+3 x^ 2 +6x+10 ,dx

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Consider the indefinite integral

I=\int \frac{x+3}{x^{2}+6x+10}\,dx.

Throughout this problem, show all algebraic steps and provide clear justification for each conclusion. (1) Rewrite the quadratic denominator in the form

(x-h)^2+k

, clearly identifying the values of

h

and

k

. Explain why this rewriting is useful for evaluating

I

. (2) Determine an explicit antiderivative for

I

. State your final answer in its simplest form. (3) Differentiate your antiderivative and verify that it reproduces the original integrand. Describe how this verification supports the correctness of your result.

Related theory

Integration Techniques
Standard algebraic integrals
Integrals of additional types
Integrals

Browse the problem library