Evaluate the integrals in Exercises 1-24 using integration by parts. ∫4 x sec^2 2 x d x

Name: Problem 14, Chapter 8: Calculus
Uploaded: 2020-08-02T08:00:19-08:00
Duration: 1 min 46 s
Channel: Darshan Maheshwari
Description: Evaluate the integrals in Exercises 1-24 using integration by parts. ∫4 x sec^2 2 x d x

step 1 Problem 14 says that we have to integrate 4x, 6 square 2x dx using integration by parts. Let's t

Evaluate the integrals in Exercises 1-24 using integration...

Key Concepts

Integration by Parts

Integration by parts is a technique used to integrate products of functions, based on the formula ? u dv = uv ? ? v du. It transforms a difficult integral into simpler parts by choosing one part of the product to differentiate (u) and another to integrate (dv), making it a fundamental tool in calculus for handling integrals that don’t easily succumb to basic integration methods.

Selection of u and dv

Choosing u and dv appropriately is crucial to effectively apply integration by parts. A common heuristic for selecting u is the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential), which helps in deciding the order in which functions are differentiated and integrated, ensuring the transformed integral is simpler than the original.

Integration of Trigonometric Functions

Integrals involving trigonometric functions, such as sec²(2x), often require standard trigonometric integration techniques. Knowing the antiderivatives of common trigonometric functions is essential; for example, the integral of sec²(?) is tan(?) plus a constant, which is fundamental when the integrand involves a trigonometric function scaled by a constant.

Application of the Chain Rule in Integration

When integrating functions like sec²(2x), it is important to account for the inner function (2x) by using the chain rule in reverse, which may require a simple substitution to handle the constant multiplier. This process ensures that the derivative of the inner function is appropriately incorporated into the antiderivative, maintaining the integrity of the integration process.

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