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Solution: Rational functions of trigonometric functions An
Chapter 4, Problem 73E(choose chapter or problem)
Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u=tan(x/2) or \(x=2 \tan ^{-1} u\). The following relations are used in making this change of variables.
A: \(d x=\frac{2}{1+u^{2}} d u\)
B: \(\sin x=\frac{2 u}{1+u^{2}}\)
C: \(\cos x=\frac{1-u^{2}}{1+u^{2}}\)
Evaluate \(\int \frac{d x}{1+\sin x}\).
Questions & Answers
QUESTION:
Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u=tan(x/2) or \(x=2 \tan ^{-1} u\). The following relations are used in making this change of variables.
A: \(d x=\frac{2}{1+u^{2}} d u\)
B: \(\sin x=\frac{2 u}{1+u^{2}}\)
C: \(\cos x=\frac{1-u^{2}}{1+u^{2}}\)
Evaluate \(\int \frac{d x}{1+\sin x}\).
ANSWER:Solution:-Step1Given thatAn integrand with trigonometric functions in the numerator and denominator can often be converted to a r