How to find the integral using long division and natural logarithms

The video tutorial explains the process of simplifying expressions using long division, focusing on dividing polynomials. It begins with setting up the division by identifying the divisor and dividend, then proceeds to divide using the first term of the divisor. The tutorial emphasizes aligning terms vertically for subtraction and handling remainders. It concludes with integrating the expression using U substitution, demonstrating how to rewrite and solve integrals involving polynomials.

What is the first step in performing long division with polynomials?

• Subtract the divisor from the dividend.

• Divide the last term of the dividend by the last term of the divisor.

• Divide the first term of the dividend by the first term of the divisor.

• Multiply the entire dividend by the divisor.

Why is it important to align terms vertically when subtracting polynomials?

• To ensure like terms are subtracted correctly.

• To simplify the multiplication process.

• To make it easier to add the terms together.

• To ensure the coefficients are multiplied correctly.

How is the remainder expressed in polynomial long division?

• As a separate polynomial.

• As a fraction over the original dividend.

• As a fraction over the divisor.

• As a decimal value.

What is the purpose of U-substitution in integration?

• To simplify the integration process by changing variables.

• To find the derivative of a function.

• To evaluate definite integrals.

• To solve differential equations.

In the context of U-substitution, what does 'DU' represent?

• The constant of integration.

• The differential of the new variable U.

• The integral of the function.

• The derivative of the original function.