Mathematics

Mathematics

Problem

Evaluate the following integral, showing every step clearly: a. (i) Obtain the indefinite integral ∫cos⁡x1+sin⁡x dx\displaystyle\int \dfrac{\cos x}{1+\sin x}\,dx. b. (ii) Hence, calculate ∫0π/4cos⁡x1+sin⁡x dx\displaystyle\int_{0}^{\pi/4} \dfrac{\cos x}{1+\sin x}\,dx.
Intuitive explanation

This integral asks you to find the antiderivative and apply the Fundamental Theorem of Calculus. The key insight is recognizing which integration rules (power rule, substitution, or special forms) apply to each term, then carefully tracking the limits through your work. Breaking the integral into simpler pieces and handling them one at a time makes even complicated-looking expressions manageable.

Intuitive explanation

This integral asks you to find the antiderivative and apply the Fundamental Theorem of Calculus. The key insight is recognizing which integration rules (power rule, substitution, or special forms) apply to each term, then carefully tracking the limits through your work. Breaking the integral into simpler pieces and handling them one at a time makes even complicated-looking expressions manageable.

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