The value of ∫−11ex+e−x(1+∣x∣−x)ex+(∣x∣−x)e−x \, dx is equal to:
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Step-by-Step Solution
Step 1: Split the integrand
First, we can split the given integral into two parts. The first part contains the ex term from the numerator, and the second part contains the remaining terms. This separation helps simplify the expression.
Step 2: Simplify the second integral
In the second integral, we can factor out (∣x∣−x) from the numerator, which simplifies the expression to (∣x∣−x)ex+e−xex+e−x=(∣x∣−x).
Step 3: Split the second integral further
We can further split the second integral into two separate integrals: one for ∣x∣ and one for x. This allows us to evaluate each part individually.
Step 4: Evaluate integrals of odd and even functions
For an integral from −a to a: if f(x) is an odd function (f(−x)=−f(x)), then ∫−aaf(x)dx=0. Since x is an odd function, ∫−11xdx=0. If f(x) is an even function (f(−x)=f(x)), then ∫−aaf(x)dx=2∫0af(x)dx. Since ∣x∣ is an even function, ∫−11∣x∣dx=2∫01xdx.
Step 5: Evaluate the remaining integrals
For the first integral, multiply the numerator and denominator by ex. Let u=e2x+1, then du=2e2xdx. The integral becomes 21∫u1du=21ln∣u∣. Evaluating from −1 to 1 gives 21[ln(e2x+1)]−11=21(ln(e2+1)−ln(e−2+1)). For the second integral, we evaluate ∫x1/2dx=3/2x3/2=32x3/2 from 0 to 1.
Step 6: Combine the results
Substitute the evaluated values back into the expression for I. The logarithmic terms simplify to 1. The term 2×32 becomes 34.
Step 7: Final Calculation
Add the simplified terms to get the final value of the integral.