Jawaban Soal Teknik Integrasi Bab 7.1 Purcell

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Jawaban Soal Teknik Integrasi Bab 7.1 Purcell

Carilah jawaban dari masalah!

1. $\int (x-2)^5 dx =$

Misal $u=x-2 \Rightarrow \frac{du}{dx}=1\Rightarrow dx=du$

$\int (x-2)^5 dx = \int u^5 du = \frac{1}{6}u^6+C = \frac{1}{6}(x-2)^6+C$

2. $\int (3x)^{\frac{1}{2}} dx =\frac{1}{2}(3x)^{-\frac{1}{2}}+C$

5. $\int \frac{1}{x^2+4} dx = \int \frac{1}{x^2+2^2} dx=\frac{1}{2}tan^{-1}(\frac{x}{2})+C$

8. $\int \frac{2t^2}{2t^2+1}=$

Catatan :

$\frac{x}{x+1}=1-\frac{1}{x+1}$

Proses :

$\int \frac{2t^2}{2t^2+1}=\int 1- \frac{1}{2t^2+1}= t- \int \frac{1}{(\sqrt{2}t)^2+1}$

Misalkan $u=\sqrt{2}t \Rightarrow \frac{du}{dt}=\sqrt{2} \Rightarrow dt=\frac{du}{\sqrt{2}}$

$t- \int \frac{1}{(\sqrt{2}t)^2+1}=t- \frac{1}{\sqrt{2}}tan^{-1}\sqrt{2}t+C$

11. $\int \frac{tan z}{cos^2z}dz =$

Misalkan $u= tan z \Rightarrow \frac{du}{dz}=\frac{1}{cos^2z}$

13. $\int \frac {sin \sqrt {t}}{\sqrt{t}} dt=$

Misalka $u= \sqrt{t}$

14. $\int \frac{2x}{\sqrt{1-x^4}}dx=$

$\int \frac{2x}{\sqrt{1-x^4}}dx=\int \frac{2x}{\sqrt{1-(x^2)^2}}dx$

Misalkan $u=x^2$

15. $\int \frac{cos x}{1+sin^2 x}dx$

Misalkan $u=sin x$

17 $\int \frac{3x^2+2x}{x+1}dx=$

Lakukan pembagian biasa sehingga diperoleh hasil 3x dengan sisa $\frac {-x}{x+1}$

Rubah $\frac {-x}{x+1}$ menjadi $\frac{1}{x+1}-1$

19. $\int \frac{sin (ln 4x^2)}{x}dx$

Misal $u=ln4x^2$

21. $\int \frac{6e^x}{\sqrt{1-e^{2x}}}dx=$

$\int \frac{6e^x}{\sqrt{1-e^{2x}}}dx=\int \frac{6e^x}{\sqrt{1-(e^{x})^2}}dx$

27. $\int \frac{sin x -cos x}{sin x}dx =$

$\int \frac{sin x -cos x}{sin x}dx =\int \frac{sin x }{sin x}- \frac{cos x}{sin x}dx$

28. $\frac{sin (4t-1)}{1-sin^2(4t-1)}dt =$

ganti $1-sin^2(4t-1)$ menjadi $cos^2(4t-1)$

32. $\int \frac{(6t-1)sin \sqrt{3t^2-t-1}}{\sqrt{3t^2-t-1}}$

Misalkan $u=3t^2-t-1$

33. $\int \frac{t^2cos (t^3-2)}{sin^2(t^3-2)}dt$

Misalkan $u=sin( t^3-2)$

34. $\int \frac{1+cos 2x }{sin^2 2x}=$

$\int \frac{1+cos 2x }{sin^2 2x}=\int \frac{1}{sin^2 2x}+ \int \frac{cos 2x}{sin^2 2x}$

37. $\int \frac{e^{tan-1 2t}}{1+4t^2}dt=$

Misal $u=tan^-1 2t$

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