Integration

is an even function, symmetric about y-axis.

Let . Let the denominator be . Then: Let numerator = derivative of numerator Then: Try differentiating option (d), you will get the integrand. Hence verified.

Let us simplify the integrand: Now integrate term-by-term: First term: This integral is known to simplify to , as confirmed by checking its derivative. Differentiating : This matches the integrand when simplified, confirming the correct result.

Use integration by parts: Let , Then, , Evaluate from 0 to 1:

Let , When Then: So: Use Beta function: Here:

Let us denote This is an odd function times an odd function over symmetric limits, thus the integrand is even. Use substitution: Let , then Use integration by parts:

The value of the integral is evaluated by considering the behavior of the greatest integer function and simplifying the expression over the given range.
The result is .

To find the area bounded by the curves, we compute the integral of from to .
The result of the integration gives the area .

The given integral simplifies after substitution and integration, leading to the value .

The integral is evaluated using standard trigonometric integral techniques, leading to the result .

We use the substitution method to solve this integral. By simplifying the expression using trigonometric identities and performing the integration, we get the solution .

The integral given is . We can use the substitution , which gives . The integral then simplifies to: Substituting back , we get:

The integral of is , so: Thus, the correct answer is , but there was an error in the transcription above, we need to correctly update the integral form.

Step 1: Use integration by parts.
Let , .
Step 2: Differentiate and integrate.

Step 3: Apply the formula.

Step 4: Compute the integral.