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lesya692 [45]
3 years ago
15

Use trigonometric substitution to evaluate the integral 13 + 12x − x2 dx . First, write the expression under the radical in an a

ppropriate form so that a trigonometric substitution can be performed. 13 + 12x − x2
Mathematics
1 answer:
SCORPION-xisa [38]3 years ago
3 0

I don't see a square root sign anywhere, so I'll assume the integral is

\displaystyle\int\sqrt{13+12x-x^2}\,\mathrm dx

First complete the square:

13+12x-x^2=49-(6-x)^2=7^2-(6-x)^2

Now in the integral, substitute

6-x=7\sin t\implies\mathrm dx=-7\cos t\,\mathrm dt

so that

t=\sin^{-1}\left(\dfrac{6-x}7\right)

Under this change of variables, we have

7^2-(6-x)^2=7^2-7^2\sin^2t=7^2(1-\sin^2t)=7^2\cos^2t

so that

\displaystyle\int\sqrt{13+12x-x^2}\,\mathrm dx=-7\int\sqrt{7^2\cos^2t}\,\cos t\,\mathrm dt=-49\int|\cos t|\cos t\,\mathrm dt

Under the right conditions, namely that cos(<em>t</em>) > 0, we can further reduce the integrand to

|\cos t|\cos t=\cos^2t=\dfrac{1+\cos(2t)}2

\displaystyle-49\int|\cos t|\cos t\,\mathrm dt=-\frac{49}2\int(1+\cos(2t))\,\mathrm dt=-\frac{49}2\left(t+\frac12\sin(2t)\right)+C

Expand the sine term as

\dfrac12\sin(2t)}=\sin t\cos t

Then

t=\sin^{-1}\left(\dfrac{6-x}7\right)\implies \sin t=\dfrac{6-x}7

t=\sin^{-1}\left(\dfrac{6-x}7\right)\implies \cos t=\sqrt{7^2-(6-x)^2}=\sqrt{13+12x-x^2}

So the integral is

\displaystyle-\frac{49}2\left(\sin^{-1}\left(\dfrac{6-x}7\right)+\dfrac{6-x}7\sqrt{13+12x-x^2}\right)+C

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