4∫01(3+x2+1+x21)dx−3loge(3) is equal to:
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Step-by-Step Solution
Step 1: Rationalize the integrand
To simplify the integrand, we rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is (3+x2−1+x2). This uses the difference of squares formula, (a+b)(a−b)=a2−b2.
Step 2: Simplify the integrand
After rationalizing, the denominator simplifies to (3+x2)−(1+x2)=3+x2−1−x2=2. We can then factor out the constant 21 from the integral.
Step 3: Apply the standard integral formula
We use the standard integral formula for ∫a2+x2dx. For the first term, a2=3, so a=3. For the second term, a2=1, so a=1.
Step 4: Evaluate the definite integral
Now we apply the integral formula to both terms and evaluate the definite integral from 0 to 1. We substitute the upper limit x=1 and subtract the value obtained by substituting the lower limit x=0.
Step 5: Calculate the value at limits
Substitute x=1 and x=0 into the integrated expression. Note that 4=2 and loge∣1∣=0.
Step 6: Simplify the integral result
Simplify the terms. We have loge(3)=2loge(3).
Step 7: Calculate the final expression
Substitute the value of I back into the original expression and simplify. Combine the logarithmic terms.