integral of ln(sqrt(x+1)+sqrt(x)) by trig sub

TL;DR

By using trigonometric substitutions and integration by parts, we can derive the solution for integrating secant and tangent functions.

Transcript

okay this is the really interesting to quote from one of my subscribers the inter-corporate Allen of school of X plus one plus square root of x I will demonstrate his solution for you guys unfortunately I could not find his email is come anymore and you know I'd apprenticed hours long time ago I just have a chance to prison is for you guys but anyw... Read More

Key Insights

• 😑 Trigonometric substitutions, such as substituting x with tangent theta, can simplify integral expressions involving secant and tangent functions.
• ☺️ The relationship between secant and tangent functions, specifically secant theta equaling the square root of x plus 1 and tangent theta equaling the square root of x, aids in integrating these functions.
• 😑 Integration by parts is a useful technique to further simplify integral expressions and integrate the resulting terms.
• 😑 The solution for integrating secant and tangent functions involves expression manipulation and the application of trigonometric identities.

Questions & Answers

Q: How can trigonometric substitutions be used to integrate secant and tangent functions?

Trigonometric substitutions involve substituting x with tangent theta and performing differentiations to simplify the integral expression before converting it back to the x-world.

Q: What is the significance of using secant and tangent together in integration?

Secant and tangent functions complement each other in integration, as they have a relationship in which secant theta equals the square root of x plus 1 and tangent theta equals the square root of x. This relationship facilitates simplification of the integral expression.

Q: What is the role of integration by parts in integrating secant and tangent functions?

Integration by parts is used to simplify the integral expression and integrate the resulting terms. The method involves differentiating the natural logarithm of secant theta plus tangent theta and integrating the product of secant theta and tangent squared theta.

Q: How does the final solution for integrating secant and tangent functions in the x-world look like?

The solution involves expressing tangent squared theta as x times the natural logarithm of secant theta (square root of x plus 1), plus tangent theta (square root of x), minus 1/2 times secant theta (square root of x plus 1) tangent theta (square root of x), plus 1/2 times the natural logarithm of secant theta (square root of x plus 1) plus a constant (C).

Summary & Key Takeaways

• The video demonstrates an approach to integrate secant and tangent functions by utilizing trigonometric substitutions in the theta world.

• By substituting x with tangent theta and performing differentiations, the expression for the integral is simplified.

• Integration by parts is then utilized to further simplify the expression and integrate the resulting terms.