Question

Clever substitution Evaluate ∫ dx/(1 + sin x + cos x) using the substitution x=2 tan⁻¹ θ. The identities sin x = 2 sin(x/2) cos(x/2) and cos x =cos²(x/2) − sin²(x/2) are helpful.

Accepted Answer

Start by applying the substitution given: let $$x = 2 	an$$^{-1}$$ 	heta. $$This means you will express $$\sin x$$, $$\cos x$$, and $$dx in $$terms of $$	heta. $$Use the half-angle identities to rewrite $$\sin x$$ and $$\cos x in $$terms of $$\sin(x/2)$$ and $$\cos(x/2). $$Given $$x = 2 	an$$^{-1}$$ 	heta$$, note that $$x/2 = 	an$$^{-1}$$ 	heta$$, so $$\sin(x/2) = \frac{	heta}{\sqrt{1 + \$$theta^2$$}}$$ and $$\cos(x/2) = \frac{1}{\sqrt{1 + \$$theta^2$$}}. $$Substitute these into the identities: $$\sin x = 2 \sin(x/2) \cos(x/2)$$ and $$\cos x = \cos^2$(x/2) - \sin^2$(x/2)$$, and simplify the expressions in terms of $$	heta. $$Next, find $$dx in $$terms of $$d	heta by $$differentiating $$x = 2 	an$$^{-1}$$ 	heta. $$Recall that $$\frac{d}{d	heta} 	an$$^{-1}$$ 	heta = \frac{1}{1 + \$$theta^2$$}$$, so $$dx = 2 \cdot \frac{1}{1 + \$$theta^2$$} d	heta. $$Rewrite the integral $$\int \frac{dx}{1 + \sin x + \cos x}$$ entirely in terms of $$	heta$$, simplify the integrand, and then integrate with respect to $$	heta.$$

Clever substitution Evaluate ∫ dx/(1 + sin x + cos x) using the substitution x=2 tan⁻¹ θ. The identities sin x = 2 sin(x/2) cos(x/2) and cos x =cos²(x/2) − sin²(x/2) are helpful.

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Key Concepts

Trigonometric Substitution

Half-Angle Identities

Integration of Rational Functions