(b) Find:
∫x2+4x+5x+3dx
(integral, quadratic, antiderivative, calculus, integration)
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Step-by-Step Solution
Step 1: Analyze the Integrand
We need to find the integral of the given rational function. The denominator is a quadratic expression x2+4x+5. Let's examine its derivative.
Step 2: Check Derivative of Denominator
The derivative of the denominator x2+4x+5 is 2x+4. We can manipulate the numerator to contain a multiple of this derivative.
Step 3: Manipulate the Numerator
We rewrite the numerator x+3 in terms of the derivative of the denominator, which is 2x+4. We observe that x+3=21(2x+4)+1. This manipulation allows us to split the integral into two simpler parts.
Step 4: Split the Integral
Now we substitute the manipulated numerator back into the integral and split it into two separate integrals. The first integral will be a logarithmic form, and the second requires completing the square in the denominator.
Step 5: Evaluate the First Integral
For the first integral, we recognize the form ∫f(x)f′(x)dx=ln∣f(x)∣+C. Here, f(x)=x2+4x+5 and f′(x)=2x+4. Therefore, the integral evaluates to 21ln∣x2+4x+5∣.
Step 6: Complete the Square for the Second Integral
For the second integral, we complete the square in the denominator x2+4x+5. We take half of the coefficient of x (which is 4/2=2), square it (22=4), and add and subtract it to the expression. This transforms x2+4x+5 into (x+2)2+1. This form is suitable for the arctangent integral formula.
Step 7: Evaluate the Second Integral
Using the formula ∫a2+x21dx=a1tan−1(ax)+C, where for our integral a=1 and x is replaced by (x+2), the second integral evaluates to tan−1(x+2).
Step 8: Combine the Results
Finally, we combine the results of the two integrals to get the complete solution. We replace C1+C2 with a single constant C for the indefinite integral.