Answer to Question #207317 in Real Analysis for adreanna

Question #207317

Calculate the following integrals by using the integration

methods.

1

a) ∫ (4π‘₯^34 + 8π‘₯^3 + 15π‘₯)𝑑π‘₯ / (√π‘₯^2 + 4x)

0


b) ∫ 𝑑π‘₯ / sin π‘₯ βˆ™ cos^2 π‘₯


1
Expert's answer
2021-07-19T08:14:47-0400

(a)


u=x+2,du=dxu=x+2, du=dx

4π‘₯4+8π‘₯3+15π‘₯=4x3(x+2)+15x4π‘₯^4 + 8π‘₯^3 + 15π‘₯=4x^3(x+2)+15x

=4u(uβˆ’2)3+15(uβˆ’2)=4u(u-2)^3+15(u-2)

=4u4βˆ’24u3+48u2βˆ’32u+15uβˆ’30=4u^4-24u^3+48u^2-32u+15u-30


=4u2(u2βˆ’4)+16u2βˆ’24u(u2βˆ’4)βˆ’96u=4u^2(u^2-4)+16u^2-24u(u^2-4)-96u

+48u2βˆ’32u+15uβˆ’30+48u^2-32u+15u-30

=4u2(u2βˆ’4)βˆ’24u(u2βˆ’4)+64(u2βˆ’4)=4u^2(u^2-4)-24u(u^2-4)+64(u^2-4)


βˆ’113u+226-113u+226




x2+4x=x2+4x+4βˆ’4=u2βˆ’4\sqrt{x^2+4x}=\sqrt{x^2+4x+4-4}= \sqrt{u^2-4}

Table of Integrals


∫u2u2βˆ’a2du=u8(2u2βˆ’a2)u2βˆ’a2\int u^2\sqrt{u^2-a^2}du=\dfrac{u}{8}(2u^2-a^2)\sqrt{u^2-a^2}

βˆ’a48ln⁑∣u+u2βˆ’a2∣+C-\dfrac{a^4}{8}\ln|u+\sqrt{u^2-a^2}|+C


∫uu2βˆ’a2du=13(u2βˆ’a2)u2βˆ’a2+C\int u\sqrt{u^2-a^2}du=\dfrac{1}{3}(u^2-a^2)\sqrt{u^2-a^2}+C

βˆ’a48ln⁑∣u+u2βˆ’a2∣+C-\dfrac{a^4}{8}\ln|u+\sqrt{u^2-a^2}|+C


∫u2βˆ’a2du=u2u2βˆ’a2\int \sqrt{u^2-a^2}du=\dfrac{u}{2}\sqrt{u^2-a^2}

βˆ’a22ln⁑∣u+u2βˆ’a2∣+C-\dfrac{a^2}{2}\ln|u+\sqrt{u^2-a^2}|+C

∫uu2βˆ’a2du=u2βˆ’a2+C\int\dfrac{u}{\sqrt{u^2-a^2}}du=\sqrt{u^2-a^2}+C


∫1u2βˆ’a2du=ln⁑∣u+u2βˆ’a2∣+C\int\dfrac{1}{\sqrt{u^2-a^2}}du=\ln|u+\sqrt{u^2-a^2}|+C


∫4u4βˆ’24u3+48u2βˆ’17uβˆ’30u2βˆ’4du\int\dfrac{4u^4-24u^3+48u^2-17u-30}{\sqrt{u^2-4}}du




=∫4u2u2βˆ’4duβˆ’βˆ«24uu2βˆ’4du=\int4u^2\sqrt{u^2-4}du-\int24u\sqrt{u^2-4}du

+∫64u2βˆ’4duβˆ’βˆ«113uu2βˆ’4du+∫226u2βˆ’4du+\int64\sqrt{u^2-4}du-\int\dfrac{113u}{\sqrt{u^2-4}}du+\int\dfrac{226}{\sqrt{u^2-4}}du

=u(u2βˆ’2)u2βˆ’4βˆ’8ln⁑∣u+u2βˆ’4βˆ£βˆ’=u(u^2-2)\sqrt{u^2-4}-8\ln|u+\sqrt{u^2-4}|-

βˆ’8(u2βˆ’4)u2βˆ’4-8(u^2-4)\sqrt{u^2-4}

+32uu2βˆ’4βˆ’128ln⁑∣u+u2βˆ’4∣+32u\sqrt{u^2-4}-128\ln|u+\sqrt{u^2-4}|

βˆ’113u2βˆ’4+226ln⁑∣u+u2βˆ’4∣+C-113\sqrt{u^2-4}+226\ln|u+\sqrt{u^2-4}|+C

=u2βˆ’4(u3βˆ’2uβˆ’8u2+32+32uβˆ’113)=\sqrt{u^2-4}(u^3-2u-8u^2+32+32u-113)

+90ln⁑∣u+u2βˆ’4∣+C+90\ln|u+\sqrt{u^2-4}|+C

=x2+4x((x+2)3βˆ’8(x+2)2+30(x+2)βˆ’81)=\sqrt{x^2+4x}((x+2)^3-8(x+2)^2+30(x+2)-81)

+90ln⁑∣x+2+x2+4x∣+C+90\ln|x+2+\sqrt{x^2+4x}|+C

∫014π‘₯4+8π‘₯3+15π‘₯x2+4xdx\displaystyle\int_{0}^{1}\dfrac{4π‘₯^4 + 8π‘₯^3 + 15π‘₯}{ \sqrt{x^2+4x}}dx

=5(27βˆ’72+90βˆ’81)+90ln⁑(3+5)=\sqrt{5}(27-72+90-81)+90\ln(3+\sqrt{5})

βˆ’90ln⁑(2)=90ln⁑(3+52)βˆ’36-90\ln(2)=90\ln(\dfrac{3+\sqrt{5}}{2})-36


(b)


∫dxsin⁑xcos⁑2x=∫(1+tan⁑2x)dxsin⁑x\int\dfrac{dx}{\sin x\cos^2 x}=\int\dfrac{(1+\tan^2 x)dx}{\sin x}

∫dxsin⁑x=∫sin⁑xdxsin⁑2x=∫sin⁑xdx1βˆ’cos⁑2x\int\dfrac{dx}{\sin x}=\int\dfrac{\sin x dx}{\sin^2 x}=\int\dfrac{\sin xdx}{1-\cos^2 x}

=12∫sin⁑xdx1βˆ’cos⁑xβˆ’12∫sin⁑xdx1+cos⁑x=\dfrac{1}{2}\int\dfrac{\sin xdx}{1-\cos x}-\dfrac{1}{2}\int\dfrac{\sin xdx}{1+\cos x}

=12ln⁑(1βˆ’cos⁑x)βˆ’12ln⁑(1+cos⁑x)+C1=\dfrac{1}{2}\ln(1-\cos x)-\dfrac{1}{2}\ln(1+\cos x)+C_1


∫tan⁑2xdxsin⁑x=∫sin⁑xdxcos⁑2x=1cos⁑x+C2\int\dfrac{\tan^2 xdx}{\sin x}=\int\dfrac{\sin x dx}{\cos^2 x}=\dfrac{1}{\cos x}+C_2

Therefore


∫dxsin⁑xcos⁑2x=12ln⁑(1βˆ’cos⁑x)βˆ’12ln⁑(1+cos⁑x)\int\dfrac{dx}{\sin x\cos^2 x}=\dfrac{1}{2}\ln(1-\cos x)-\dfrac{1}{2}\ln(1+\cos x)

+1cos⁑x+C+\dfrac{1}{\cos x}+C


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