Trigonometric Substitutions in Integrals

This video tutorial covers integration using trigonometric substitution, focusing on the form a^2 + x^2. It begins with a review of previous concepts and introduces the substitution of x = a tan(theta). An example problem is solved step-by-step, demonstrating the process of substitution, simplification, and integration. The video concludes with back substitution to express the solution in terms of x and a preview of the next topic.

What substitution is used for integrals involving the square root of a squared minus x squared?

• x = a secant theta

• x = a tangent theta

• x = a cosine theta

• x = a sine theta

For the form square root of a squared plus x squared, what substitution should be made?

• x = a tangent theta

• x = a secant theta

• x = a cosine theta

• x = a sine theta

In the example problem, what is the value of 'a' when the integral involves the square root of four plus x squared?

• 1

• 4

• 3

• 2

What is the expression for dx when x = 2 tangent theta?

• 2 sine squared theta d theta

• 2 cosine squared theta d theta

• 2 secant squared theta d theta

• 2 tangent squared theta d theta

What trigonometric identity is used to simplify the integral in the example?

• sine squared theta + cosine squared theta = 1

• sine theta = opposite/hypotenuse

• tangent squared theta + 1 = secant squared theta

• 1 + cotangent squared theta = cosecant squared theta

What substitution is made to simplify the integral in terms of u?

• u = secant theta

• u = sine theta

• u = tangent theta

• u = cosine theta

After integration, what is the expression for the antiderivative in terms of u?

• 1/4 u^-2 + C

• -1/4 u^2 + C

• -1/4 u^-1 + C

• 1/4 u^2 + C

How is sine theta expressed in terms of x in the final answer?

• x over the square root of four plus x squared

• x over two

• two over x

• the square root of four plus x squared over x

What is the final form of the antiderivative in terms of x?

• The square root of four plus x squared all over four x plus C

• Negative the square root of four plus x squared all over four x plus C

• x over the square root of four plus x squared plus C

• Negative x over the square root of four plus x squared plus C

What will be discussed in the next video?

• The form square root of x squared minus a squared

• Integration by parts

• Integration with partial fractions

• The form square root of a squared minus x squared