Integration by Trig Substitution

Front

What is the purpose of trigonometric substitution in integration?

Back

Trigonometric substitution is used to simplify integrals involving square roots of quadratic expressions by substituting a trigonometric function for a variable, making the integral easier to evaluate.

Front

What are the common trigonometric substitutions for integrals involving sqrt(a^2 - x^2)?

Back

For integrals involving sqrt(a^2 - x^2), the common substitution is x = a sin(θ), which transforms the integral into a trigonometric form.

Front

What are the common trigonometric substitutions for integrals involving sqrt(a^2 + x^2)?

Back

For integrals involving sqrt(a^2 + x^2), the common substitution is x = a tan(θ), which simplifies the integral.

Front

What are the common trigonometric substitutions for integrals involving sqrt(x^2 - a^2)?

Back

For integrals involving sqrt(x^2 - a^2), the common substitution is x = a sec(θ), which helps in simplifying the integral.

Front

How do you determine the appropriate triangle for a given trigonometric substitution?

Back

To determine the appropriate triangle, use the substitution to identify the relationships between the sides of the triangle based on the chosen trigonometric function (e.g., opposite, adjacent, hypotenuse).

Front

What is the relationship between the sides of a right triangle and trigonometric functions?

Back

In a right triangle, the sine, cosine, and tangent functions relate the angles to the ratios of the lengths of the sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent.

Tags

CCSS.HSG.SRT.C.6

Front

What is the integral of sin(θ) dθ?

Back

The integral of sin(θ) dθ is -cos(θ) + C, where C is the constant of integration.

Front

What is the integral of cos(θ) dθ?

Back

The integral of cos(θ) dθ is sin(θ) + C, where C is the constant of integration.

Front

What is the integral of sec^2(θ) dθ?

Back

The integral of sec^2(θ) dθ is tan(θ) + C, where C is the constant of integration.

Front

What is the integral of tan(θ) dθ?

Back

The integral of tan(θ) dθ is -ln|cos(θ)| + C, where C is the constant of integration.

Tags

CCSS.HSF.TF.B.7