Latihan Matematika Wajib Kelas XI Integral Fungsi Aljabar
# 1
Pilgan

∫abxndx=...\int \frac{a}{bx^n}dx=...

A

a(n−1)bxn−1+C\frac{a(n-1)}{bx^{n-1}}+C

B

ab(1−n)xn−1+C\frac{a}{b(1-n)x^{n-1}}+C

C

anbxn−1+C\frac{an}{bx^{n-1}}+C

D

abnxn+1+C\frac{a}{bnx^{n+1}}+C

E

abnx1−n+C\frac{a}{bnx^{1-n}}+C

Pembahasan:

Ingat bahwa 1xn=x−n\frac{1}{x^n}=x^{-n} maka:

∫abxndx=∫abx−ndx\int\frac{a}{bx^n}dx=\int\frac{a}{b}x^{-n}dx


Untuk f(x)=axn, n≠−1f\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

Sehingga didapatkan:

∫abxndx\int\frac{a}{bx^n}dx

=∫abx−ndx=\int\frac{a}{b}x^{-n}dx

=ab(−n+1)x(−n+1)+C=\frac{a}{b(-n+1)}x^{(-n+1)}+C; ingat bahwa xn=1x−nx^n=\frac{1}{x^{-n}} maka:

=ab(−n+1)x−(−n+1)+C=\frac{a}{b(-n+1)x^{-(-n+1)}}+C; dikombinasikan (−n+1)=(1−n)\left(-n+1\right)=\left(1-n\right) maka:

=ab(1−n)x−(−n+1)+C=\frac{a}{b(1-n)x^{-(-n+1)}}+C

=ab(1−n)xn−1+C=\frac{a}{b(1-n)x^{n-1}}+C

Jadi, ∫abxndx=ab(1−n)xn−1+C\int\frac{a}{bx^n}dx=\frac{a}{b(1-n)x^{n-1}}+C

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