Understanding U-Substitution in Integration

This video tutorial provides practice on identifying when to use u-substitution in integration. It begins with an introduction to the concept, followed by two examples. The first example demonstrates integrating the natural log of X raised to the 10th power using u-substitution. The second example involves rewriting the tangent function in terms of sine and cosine to apply u-substitution effectively. The video emphasizes recognizing a function and its derivative to make the substitution process easier.

What is the primary goal of using u-substitution in integration?

• To evaluate definite integrals

• To find the derivative of a function

• To simplify the integrand by changing variables

• To solve differential equations

In the integral of natural log of X to the 10th power over X, what is chosen as 'u'?

• X to the 10th power

• X

• 1 over X

• Natural log of X

What is the derivative of the natural log of X?

• X

• 1 over X

• Natural log of X

• X squared

Why is the natural log of X chosen as 'u' in the given integral?

• Because it simplifies to zero

• Because it is a constant

• Because its derivative is present in the integrand

• Because it is the simplest function

How can the tangent function be rewritten to apply u-substitution?

• As sine squared of X

• As cosine of X over sine of X

• As sine of X over cosine of X

• As sine of X times cosine of X

What is the derivative of cosine of X?

• Negative cosine of X

• Cosine of X

• Negative sine of X

• Sine of X

In the integral of tangent X, what is chosen as 'u' after rewriting?

• X

• Cosine of X

• Tangent of X

• Sine of X

What is the simplified form of the integral after applying u-substitution to tangent X?

• Integral of sine of X

• Integral of u squared

• Integral of cosine of X

• Integral of 1 over u

What is the final step after evaluating the integral using u-substitution?

• Add a constant of integration

• Multiply by a constant

• Back substitute the original variable

• Differentiate the result

Why is it necessary to back substitute after finding the antiderivative?

• To simplify the expression

• To express the result in terms of the original variable

• To eliminate constants

• To verify the solution