1. ∫│x^2.sin x│dx batasnya –π/2 sampai π/2 2. ∫(arc(cos t))^2 dt batasnya ½ sampai 1 3.∫(ln x)^2 dx .... Question from @hanamutiadewi

September 2018 1 73 Report

1. ∫│x^2.sin x│dx batasnya –π/2 sampai π/2

2. ∫(arc(cos t))^2 dt batasnya ½ sampai 1

3.∫(ln x)^2 dx batasnya 1 sampai e

Takamori37 Nomor 1.
$\int_{-\pi/2}^{\pi/2}|x^2 \sin x|\, dx$
Menggunakan sifat kesimetrisan:
$2\int_0^{\pi/2}x^2 \sin x\, dx \\ $Integral parsial:$ \\ U=x^2~~~~~~~~~~dU=2x\, dx \\ dV=\sin x\, dx~~~V=-\cos x \\ 2\int_0^{\pi/2}x^2 \sin x\, dx=2(-x^2\cos x]_0^{\pi/2}-\int_0^{\pi/2}-2x\, \cos x\, dx) \\ 2\int_0^{\pi/2}x^2 \sin x\, dx=2(-x^2\cos x]_0^{\pi/2}+\int_0^{\pi/2}2x\, \cos x\, dx)$

$\\ $Pada \int_0^{\pi/2}2x\, \cos x\, dx \\ U=2x~~~~~~~~~~dU=2\, dx \\ dV=\cos x\, dx~~~V=\sin x \\ \int_0^{\pi/2}2x\, \cos x\, dx=2x\sin x]_0^{\pi/2}-\int_0^{\pi/2} 2\sin x\, dx \\ \int_0^{\pi/2}2x\, \cos x\, dx=2x\sin x+2\cos x]_0^{\pi/2}$

Sehingga:
$2\int_0^{\pi/2}x^2 \sin x\, dx=2(-x^2\cos x+2x\sin x+2\cos x]_0^{\pi/2}) \\ 2\int_0^{\pi/2}x^2 \sin x\, dx=2((-\frac{\pi^2}{4}\times 0+\pi \times 1+2\times 0)-(0+0+2)) \\ 2\int_0^{\pi/2}x^2 \sin x\, dx=2(\pi-2)$

Nomor 2.
$\int (\cos^{-1}t)^2\, dt \\ U=(\cos^{-1}t)^2~~~~~~dU=-\frac{2\cos^{-1}t}{\sqrt{1-t^2}}\, dt \\ dV=\, dt~~~~~~~~~~~~~~V=t \\ \int (\cos^{-1}t)^2\, dt=t(\cos^{-1}t)^2+\int\frac{2\cos^{-1}t}{\sqrt{1-t^2}}\, dt \\ \int (\cos^{-1}t)^2\, dt=t(\cos^{-1}t)^2-2(\sqrt{1-t^2}\cos^{-1}t+t)} \\ (1(\cos^{-1}1)^2-2(\sqrt{1-1^2}\cos^{-1}1+1)-\# \\ (1.0^2-2(0.0+1))-(\frac{1}{2}(\cos^{-1}\frac{1}{2})^2-2(\sqrt{1-\frac{1}{2}^2}\cos^{-1}\frac{1}{2}+\frac{1}{2})$

$(0-2)-(\frac{1}{2}\times\frac{\pi^2}{9}-2(\frac{1}{2}\sqrt{3}+\frac{1}{2})) \\ -2-(\frac{\pi^2}{18}-\sqrt{3}-1) \\ -1-\sqrt{3}-\frac{\pi^2}{18}$

Nomor 3.
$\int_1^e\ln^2 x\, dx \\ U=\ln^2x~~~~~dU=\frac{2}{x}\ln x\, dx \\ dV=\dx~~~~~~~~~~~V=x \\ \int_1^e\ln^2 x\, dx=x\ln^2 x]_1^e-\int_1^e2\ln x\, dx \\ \int_1^e\ln^2 x\, dx=x\ln^2 x-(x\ln x-x)]_1^e \\ \int_1^e\ln^2 x\, dx=(e\ln^2 e-(e\ln e-e))-(1\ln^21-(1\ln 1-1)) \\ \int_1^e\ln^2 x\, dx=(e-e+e)-(0-(0-1)) \\ \int_1^e\ln^2 x\, dx=e-1$

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