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Q 28.

Expert-verifiedFound in: Page 429

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Solve the integral: $\int x\mathrm{sin}xdx$

The required answer is $-x\mathrm{cos}x+\mathrm{sin}x+c$.

We have given integral is $\int x\mathrm{sin}xdx$.

We have,

$u=x\phantom{\rule{0ex}{0ex}}du=dx$

and

localid="1648636397500" $dv=\mathrm{sin}xdx\phantom{\rule{0ex}{0ex}}v=\int \mathrm{sin}xdx\phantom{\rule{0ex}{0ex}}v=-\mathrm{cos}x$

The formula of integration by parts is localid="1648645366461" $\int udv=uv-\int vdu$

$\int x\mathrm{sin}xdx\phantom{\rule{0ex}{0ex}}=x(-\mathrm{cos}x)-\int (-\mathrm{cos}x)dx\phantom{\rule{0ex}{0ex}}=-x\mathrm{cos}x+\int \mathrm{cos}x\phantom{\rule{0ex}{0ex}}=-x\mathrm{cos}x+\mathrm{sin}x+c$

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