Prove the integration formula ∫tanx d x=ln|secx|+C. (a) by using algebra and integration by subs

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Prove the integration formula ∫tanx d x=ln|secx|+C. (a) by...

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in calculus that allow the rewriting and simplification of expressions involving trigonometric functions. They enable one to express complex trigonometric integrals in forms that are more amenable to standard integration techniques, such as expressing one trigonometric function in terms of another.

Algebraic Manipulation

Algebraic manipulation involves rewriting and rearranging the integrands using fundamental algebraic operations. This process can simplify an integral by breaking it into simpler parts or by aligning it with known integral forms, making it easier to apply advanced integration techniques effectively.

Integration by Substitution

Integration by substitution is a method used to evaluate integrals by changing variables. This technique transforms an integral into a simpler one by letting a portion of the integrand be a new variable, effectively reversing the chain rule of differentiation, and thus enabling easier calculation of the antiderivative.

Chain Rule in Differentiation

The chain rule is a fundamental differentiation technique used to differentiate composite functions. It plays a crucial role when verifying antiderivatives by differentiating a result that involves composite functions, particularly those with outer functions like logarithms and inner functions involving trigonometric expressions.

Verification by Differentiation

Verification by differentiation is an application of the Fundamental Theorem of Calculus. After finding an antiderivative, differentiating it can confirm its correctness. This technique ensures that the derived antiderivative, when differentiated, recovers the original integrand, providing confidence in the integration process.

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