Day: October 19, 2023

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202310191146 Exercise 7.9.26

Evaluate the integral

\displaystyle{\int x\sin x\cos x\,\mathrm{d}x}.

Extracted from James Stewart. (2012). Calculus (8e)

Roughwork.

(abortive attempt)

\begin{aligned} & \quad\enspace \int x\sin x\cos x\,\mathrm{d}x \\ & = \int x\sin x\cos x \bigg(\frac{\mathrm{d}(\sin x)}{\cos x}\bigg) \\ & = \int x\sin x\,\mathrm{d}(\sin x) \\ & = (x\sin x)(\sin x) - \int (\sin x)\bigg(\frac{\mathrm{d}(x\sin x)}{\mathrm{d}x}\bigg)\,\mathrm{d}x + C\quad\textrm{for some constant} \\ & = x\sin^2x - \int \sin x\,\mathrm{d}(x\sin x) \\ & = x\sin^2x - \bigg\{ (\sin x)(x\sin x) - \int (x\sin x)\bigg(\frac{\mathrm{d}}{\mathrm{d}x}(\sin x)\bigg) \,\mathrm{d}x \bigg\} \\ & = x\sin^2x - \bigg\{ x\sin^2x - \int x\sin x\cos x\,\mathrm{d}x \bigg\} \\ & = \int x\sin x\cos x\,\mathrm{d}x \\ \end{aligned}

As a last resort, I need help from the following

Fundamental Theorem of Calculus. Let f be a continuous real-valued function defined on a closed interval [a,b], Let F be the function defined, for all x in [a,b], by

\displaystyle{F(x)=\int_{a}^{x}f(t)\,\mathrm{d}t}

Then F is uniformly continuous on [a,b] and differentiable on the open interval (a,b), and

F'(x)=f(x)

for all x in (a,b) so F is an antiderivative of f.

Cited from Wikipedia on Fundamental theorem of calculus

Let \displaystyle{\frac{\mathrm{d}F(x)}{\mathrm{d}x}=x\sin x\cos x}.

But for using foresight, one should not have written

\begin{aligned} F(x) & :=\frac{x}{2}\sin^2x+G(x) \\ \frac{\mathrm{d}F(x)}{\mathrm{d}x} & = x\sin x\cos x + \frac{\sin^2x}{2} + G'(x) \\ G'(x) & := -\frac{\sin^2x}{2} \\ G(x) & = \int G'(x)\,\mathrm{d}x \\ & = -\frac{1}{2}\int\sin^2x\,\mathrm{d}x \\ & = -\frac{1}{2}\int \bigg( \frac{1-\cos 2x}{2}\bigg) \,\mathrm{d}x\\ & = -\frac{1}{4}\bigg\{ \int \mathrm{d}x - \int \cos 2x\,\mathrm{d}x \bigg\} \\ & = -\frac{1}{4}\bigg\{ x - \frac{\sin 2x}{2} \bigg\} \\ F(x) & = \frac{x}{2}\sin^2x - \frac{1}{4}\bigg( x - \frac{\sin 2x}{2} \bigg) \\ & = \frac{x\sin^2x}{2} + \frac{\sin 2x}{8} - \frac{x}{4} \\ \end{aligned}

with some constant C.

The case is closed.