Bank Soal Matematika SMA Integral Fungsi Aljabar

Soal

Pilgan

Hasil dari (4x5+2(x1)42x3)dx\int(4x^5+2\left(x-1\right)^4-2\sqrt{x^3})dx adalah ....

A

23x6+25(x1)545x5+C\frac{2}{3}x^6+\frac{2}{5}\left(x-1\right)^5-\frac{4}{5}\sqrt{x^5}+C

B

23x6+25(x1)525x5+C\frac{2}{3}x^6+\frac{2}{5}\left(x-1\right)^5-\frac{2}{5}\sqrt{x^5}+C

C

23x6+2(x1)525x5+C\frac{2}{3}x^6+2\left(x-1\right)^5-\frac{2}{5}\sqrt{x^5}+C

D

23x6+25(x1)54x5+C\frac{2}{3}x^6+\frac{2}{5}\left(x-1\right)^5-4\sqrt{x^5}+C

E

43x6+25(x1)545x+C\frac{4}{3}x^6+\frac{2}{5}\left(x-1\right)^5-\frac{4}{5}\sqrt{x}+C

Pembahasan:

Untuk f(x)=axn, n1f\left(x\right)=ax^n,\ n\ne-1 maka:

axndx=an+1xn+1+C\int ax^ndx=\frac{a}{n+1}x^{n+1}+C

Integral tersebut terdiri dari beberapa integral yang dijumlahkan dan dikurangkan, maka kita uraikan terlebih dahulu dengan menggunakan aturan Integral Penjumlahan dan Pengurangan, yaitu:

[f(x)±g(x)]dx=f(x)dx±g(x)dx\int\left[f\left(x\right)\pm g\left(x\right)\right]dx=\int f\left(x\right)dx\pm\int g\left(x\right)dx

(4x5+2(x1)42x3)dx=4x5dx+2(x1)4dx2x3dx\int(4x^5+2\left(x-1\right)^4-2\sqrt{x^3})dx=\int4x^5dx+\int2\left(x-1\right)^4dx-\int2\sqrt{x^3}dx


Ingat bahwa nxm=xmn^n\sqrt{x^m}=x^{\frac{m}{n}}, maka:

4x5dx+2(x1)4dx2x3dx=4x5dx+2(x1)4dx2(x32)dx\int4x^5dx+\int2\left(x-1\right)^4dx-\int2\sqrt{x^3}dx=\int4x^5dx+\int2\left(x-1\right)^4dx-\int2\left(x^{\frac{3}{2}}\right)dx


Selanjutnya (ax+b)ndx=1a(n+1)(ax+b)n+1+C\int\left(ax+b\right)^ndx=\frac{1}{a\left(n+1\right)}\left(ax+b\right)^{n+1}+C maka:

4x5dx+2(x1)4dx2(x32)dx=4x5dx+2(x1)4dx2(x32)dx\int4x^5dx+\int2\left(x-1\right)^4dx-\int2\left(x^{\frac{3}{2}}\right)dx=\int4x^5dx+2\int\left(x-1\right)^4dx-\int2\left(x^{\frac{3}{2}}\right)dx

=45+1x5+1+2(11(4+1)(x1)4+1)232+1x32+1+C=\frac{4}{5+1}x^{5+1}+2\left(\frac{1}{1\left(4+1\right)}\left(x-1\right)^{4+1}\right)-\frac{2}{\frac{3}{2}+1}x^{\frac{3}{2}+1}+C

=46x6+2(15(x1)5)232+22x32+22+C=\frac{4}{6}x^6+2\left(\frac{1}{5}\left(x-1\right)^5\right)-\frac{2}{\frac{3}{2}+\frac{2}{2}}x^{\frac{3}{2}+\frac{2}{2}}+C

=23x6+25(x1)5252x52+C=\frac{2}{3}x^6+\frac{2}{5}\left(x-1\right)^5-\frac{2}{\frac{5}{2}}x^{\frac{5}{2}}+C

=23x6+25(x1)545x5+C=\frac{2}{3}x^6+\frac{2}{5}\left(x-1\right)^5-\frac{4}{5}\sqrt{x^5}+C


Jadi, hasil integral fungsi tersebut adalah 23x6+25(x1)545x5+C\frac{2}{3}x^6+\frac{2}{5}\left(x-1\right)^5-\frac{4}{5}\sqrt{x^5}+C

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