Integration

Step 1: Use a trigonometric identity.
We use the identity to simplify the integral.
Step 2: Conclusion.
After simplifying and performing the integration, we find that the value of the integral is 2.
Final Answer:

Step 1: Solve the integral.
Use the substitution to solve the integral.
Step 2: Conclusion.
Thus, the solution to the integral is .
Final Answer:

We are given the integral: Since changes sign at , we split the integral at . Thus, we write: Step 1: Evaluate the first integral.
For , is negative, so we have: Integrating:
Step 2: Evaluate the second integral.
For , is positive, so we have: Integrating: Simplifying:
Step 3: Add the results.
Now, add the results of the two integrals: Thus, the value of the integral is , and the correct answer is (b).

We are given the integral: Step 1: Substitution to simplify the integral.
We make the substitution , where , and , . Then the integral becomes:
Step 2: Evaluate the definite integral and substitute the limits.
Now the integral can be evaluated using known standard methods, and it is found to simplify to:
Step 3: Conclusion.
Thus, the value of the integral is , and the correct answer is (b).

We are asked to integrate the following expression: Step 1: Simplifying the expression.
First, we simplify the expression by recognizing a standard trigonometric identity. Start by rewriting in a form that makes it easier to integrate. We can express as follows: Thus, the original integral becomes:
Step 2: Substitution.
Now, factor out from the denominator: Next, perform the substitution , which leads to . This transforms the integral into:
Step 3: Applying standard integral.
The integral of the form is a standard result: Substitute back into the equation:
Step 4: Conclusion.
Thus, the integral evaluates to: Therefore, the correct answer is (c).

Step 1: Simplifying the expression.
The integral is solved by breaking down the expression and using integration by parts. After solving, we find that the result is .

Step 2: Conclusion.
The integral evaluates to , corresponding to option (3).

Step 1: Understanding the function.
The function is the fractional part of , which is periodic with a period of 1. The integral over one period results in 0.

Step 2: Conclusion.
The value of the integral is 0, corresponding to option (1).

We compute the integral using standard logarithmic and trigonometric identities.

To find the area bounded by the curves, we integrate the difference between the two functions over the specified interval.

Use substitution methods to solve the integral and simplify it.

To solve the given integral, we use substitution and apply integration rules for powers of functions. The integral evaluates to .
Step 2: Conclusion.
The correct answer is (B).

The given integral needs to be evaluated using integration by parts. The solution requires splitting the integral into simpler parts.
Final Answer:

Step 1: Set up the integral.
The area can be calculated by integrating from to .
Step 2: Conclusion.
Thus, the area bounded by the given curves is .
Final Answer:

Step 1: Use a standard integral.
This is a standard integral, and its value can be derived from integral tables or computed using substitution and symmetry.
Step 2: Conclusion.
Thus, the value of the integral is .
Final Answer:

Step 1: Integrate the function.
Using the given function , we can integrate it from 0 to .
Step 2: Conclusion.
Thus, the integral of over the given interval is 1.
Final Answer:

Step 1: The greatest integer function, , takes the value of the greatest integer less than or equal to . The integral is split into two parts: Step 2: After evaluating, we find the result of the integral is 0.

Final Answer:

Step 1: Use the reduction formula or trigonometric identity to simplify .
Step 2: After simplification and integration, we obtain .

Final Answer:

Step 1: To find the area between the curve and the lines, we need to set up an integral. The equation of the curve is . The bounds for are from to .
Step 2: The area is given by: Step 3: Solving the integral, we get an area of 6 sq. units.

Final Answer:

Step 1: Break the integrand into partial fractions, where we need to find the value of .
Step 2: Use the method of partial fractions to decompose into two terms: Step 3: By comparing the coefficients, we find that .

Final Answer:

We break the integral into pieces based on the absolute value function, and then evaluate each part to obtain the final result.
Final Answer:

Use standard integration techniques and simplify the integral to find the value as .
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Using substitution and integration techniques, we simplify the integral and find the result as .
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The absolute value function requires breaking the integral into parts where and have different signs. The total area is calculated as .
Final Answer:

By differentiating the given equation and comparing it with the integrand, we find that and .
Final Answer:

Differentiate to verify.

This is the standard Gaussian integral: