Editorial - Matlan - Latihan Integral 1

Problem 1

the integral of open paren 2 x plus 1 close paren d of x equals 2 x squared over 2 plus one oneth x plus C equals x squared plus x plus C

Problem 2

the integral of open paren x squared minus x plus 3 close paren d of x equals one third x cubed minus the fraction with numerator x squared and denominator 2 plus 3 x plus C

Problem 3

the integral of open paren the fraction with numerator 1 and denominator x to the fourth power plus the fraction with numerator 1 and denominator x raised to the 1 divided by 4 power close paren d of x equals the integral of open paren x to the negative 4 power plus x raised to the negative 1 divided by 4 power close paren d of x equals negative one third x to the negative 3 power plus four thirds x raised to the 3 divided by 4 power plus C

Problem 4

the integral of open paren x squared times open paren x plus 2 close paren close paren d of x equals the integral of open paren x cubed plus 2 x squared close paren d of x equals one fourth x to the fourth power plus two thirds x cubed plus C

Problem 5

the integral of open paren open paren x squared plus 3 close paren squared close paren d of x equals the integral of open paren x to the fourth power plus 6 x squared plus 9 close paren d of x equals one fifth x to the fifth power plus 2 x cubed plus 9 x plus C

Problem 6

the integral of open paren open paren x cubed minus 2 x close paren squared close paren d of x equals the integral of open paren x to the sixth power minus 4 x to the fourth power plus 4 x squared close paren d of x equals one seventh x to the seventh power minus four fifths x to the fifth power plus four thirds x cubed plus C

Problem 7

the integral of open paren the fraction with numerator x to the fifth power minus x squared plus 1 and denominator x squared close paren d of x equals the integral of open paren x cubed minus 1 plus x to the negative 2 power close paren equals one fourth x to the fourth power minus x minus x to the negative 1 power plus C

Problem 8

the integral of open paren 10 over x the fifth root of x squared close paren d of x equals the integral of open paren the fraction with numerator 10 and denominator x times x raised to the 2 divided by 5 power close paren d of x equals the integral of open paren the fraction with numerator 10 and denominator x raised to the 7 divided by 5 power close paren d of x equals the integral of open paren 10 x raised to the negative 7 divided by 5 power close paren d of x

10 x raised to the negative 2 divided by 5 power over negative two fifths plus C equals negative five halves times 10 times x raised to the negative 2 divided by 5 power plus C equals negative the fraction with numerator 25 and denominator x raised to the 2 divided by 5 power plus C

Problem 9

the integral of open paren the fraction with numerator 6 x squared plus 4 the square root of x and denominator x close paren d of x equals the integral of open paren 6 x plus 4 x raised to the negative 1 divided by 2 power close paren d of x

3 x squared plus 8 x raised to the 1 divided by 2 power plus C

Problem 10

the integral of open paren the fraction with numerator 9 the cube root of x plus 8 the ninth root of x squared plus 10 x and denominator the cube root of x close paren d of x equals the integral of open paren 9 plus 8 x raised to the negative 1 divided by 9 power plus 10 x raised to the 2 divided by 3 power close paren d of x

9 x plus 9 x raised to the 8 divided by 9 power plus 6 x raised to the 5 divided by 3 power plus C

Problem 11

the integral of open paren open paren x minus 1 close paren squared times open paren x plus 3 close paren over x the square root of x close paren d of x equals the integral of open paren the fraction with numerator x cubed plus x squared minus 5 x plus 3 and denominator x raised to the 3 divided by 2 power close paren d of x

the integral of open paren open paren x raised to the 3 divided by 2 power plus x raised to the 1 divided by 2 power minus 5 x raised to the negative 1 divided by 2 power plus 3 x raised to the negative 3 divided by 2 power close paren close paren d of x

two fifths x raised to the 5 divided by 2 power plus two thirds x raised to the 3 divided by 2 power minus 10 x raised to the 1 divided by 2 power minus 6 x raised to the negative 1 divided by 2 power plus C

Problem 12

the integral of open paren the fraction with numerator open paren x squared minus x to the negative 1 power close paren squared and denominator the square root of x close paren d of x

the integral of open paren the fraction with numerator x to the fourth power minus 2 x plus x to the negative 2 power and denominator x raised to the 1 divided by 2 power close paren d of x

the integral of open paren x raised to the 7 divided by 2 power minus 2 x raised to the 1 divided by 2 power plus x raised to the negative 5 divided by 2 power close paren d of x

two ninths x raised to the 9 divided by 2 power minus four thirds x raised to the 3 divided by 2 power minus two thirds x raised to the negative 3 divided by 2 power plus C

Problem 13

f with hook prime of x equals 12 times open paren x plus 1 close paren times open paren x plus 2 close paren

f with hook of negative 3 equals negative 18

f with hook of x equals question mark

Sol.

f with hook prime of x equals 12 times open paren x squared plus 3 x plus 2 close paren

f with hook of x equals the integral of open paren 12 times open paren x squared plus 3 x plus 2 close paren close paren d of x equals 12 times open paren one third x cubed plus three halves x squared plus 2 x plus C close paren

f with hook of negative 3 equals 12 times open paren one third times negative 27 plus three halves times 9 plus 2 times negative 3 plus C close paren equals negative 18 plus 12 C

12 C equals negative 18 plus 18 equals 0

therefore C equals 0

f with hook of x equals 12 times open paren one third x cubed plus three halves x squared plus 2 x plus 0 close paren equals 4 x cubed plus 18 x squared plus 24 x

Problem 14

y equals f with hook of x

f with hook prime of x equals 3 x squared plus 2

f with hook of 2 equals 5

f with hook of x equals question mark

Sol.

f with hook of x equals the integral of open paren 3 x squared plus 2 close paren d of x equals x cubed plus 2 x plus C

f with hook of 2 equals 8 plus 4 plus C equals 5

C equals 5 minus 8 minus 4 equals negative 7

therefore f with hook of x equals x cubed plus 2 x minus 7

Problem 15

Gradien garis singgung di open paren x comma y close paren adalah open paren 2 x plus 3 the square root of x close paren

Kurva melalui open paren 4 comma 2 close paren

Cari garis singgung di x equals 1

Sol.

f with hook of 4 equals 2

f with hook prime of x equals 2 x plus 3 the square root of x

Find gradient of tangent at x equals 1

f with hook prime of 1 equals 2 plus 3 equals 5

therefore m equals 5

Find f with hook of x

f with hook of x equals x squared plus 2 x raised to the 3 divided by 2 power plus C

f with hook of 4 equals 16 plus 2 times 4 raised to the 3 divided by 2 power plus C equals 32 plus C equals 2

therefore C equals negative 30

Find f with hook of 1

f with hook of 1 equals 1 plus 2 minus 30 equals negative 27

Tangent is at open paren 1 comma negative 27 close paren

y equals m x plus b

y equals 5 x plus b

negative 27 equals 5 plus b

b equals negative 32

therefore y equals 5 x minus 32