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Tentukan nilai dari: c.

Pertanyaan

Tentukan nilai dari:

c. integral open parentheses 5 e to the power of x plus 1 third x cubed minus 4 over x close parentheses text  dx end text 

Pembahasan Soal:

Rumus integral tak tentu:

integral a x to the power of n space d x equals fraction numerator a over denominator n plus 1 end fraction x to the power of n plus 1 end exponent plus c

integral 1 over x space d x equals ln space open vertical bar x close vertical bar plus c

integral e to the power of x space d x equals e to the power of x plus c

Integral pada soal di atas dapat ditentukan sebagai berikut.

integral open parentheses 5 e to the power of x plus 1 third x cubed minus 4 over x close parentheses space d x equals 5 e to the power of x plus 1 third times 1 fourth x to the power of 4 minus 4 space ln space open vertical bar x close vertical bar plus c equals 5 e to the power of x plus 1 over 12 x to the power of 4 minus 4 space ln space open vertical bar x close vertical bar plus c

Dengan demikian, integral open parentheses 5 e to the power of x plus 1 third x cubed minus 4 over x close parentheses space d x equals 5 e to the power of x plus 1 over 12 x to the power of 4 minus 4 space ln space open vertical bar x close vertical bar plus c 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

H. Eka

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 05 Juni 2021

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Pertanyaan yang serupa

Tentukan nilai dari: a.

Pembahasan Soal:

Rumus integral tak tentu:

integral a x to the power of n space d x equals fraction numerator a over denominator n plus 1 end fraction x to the power of n plus 1 end exponent plus c

integral 1 over x space d x equals ln space open vertical bar x close vertical bar plus c

Integral pada soal di atas dapat ditentukan sebagai berikut.

integral open parentheses x squared plus 3 over x close parentheses space d x equals 1 third x cubed plus 3 space ln space open vertical bar x close vertical bar plus c

Dengan demikian, integral open parentheses x squared plus 3 over x close parentheses space d x equals 1 third x cubed plus 3 space ln space open vertical bar x close vertical bar plus c 

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Roboguru

Tentukan nilai dari: b.

Pembahasan Soal:

Rumus integral tak tentu:

integral a x to the power of n space d x equals fraction numerator a over denominator n plus 1 end fraction x to the power of n plus 1 end exponent plus c

integral 1 over x space d x equals ln space open vertical bar x close vertical bar plus c

integral e to the power of x space d x equals e to the power of x plus c

Integral pada soal di atas dapat ditentukan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell integral open parentheses fraction numerator 1 over denominator 2 x end fraction plus x squared minus e to the power of x close parentheses space d x end cell row blank equals cell integral open parentheses 1 half times 1 over x plus x squared minus e to the power of x close parentheses space d x end cell row blank equals cell 1 half space ln space open vertical bar x close vertical bar plus 1 third x cubed minus e to the power of x plus c end cell end table

Dengan demikian, integral open parentheses fraction numerator 1 over denominator 2 x end fraction plus x squared minus e to the power of x close parentheses space d x equals 1 half space ln space open vertical bar x close vertical bar plus 1 third x cubed minus e to the power of x plus c 

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Carilah !

Pembahasan Soal:

Dengan menggunakan sifat integral, maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator 3 over denominator x squared minus 1 end fraction d x end cell equals cell 3 times open parentheses integral fraction numerator 1 over denominator x squared minus 1 end fraction d x close parentheses end cell row blank equals cell 3 times open parentheses integral fraction numerator 1 over denominator negative left parenthesis negative x squared plus 1 right parenthesis end fraction d x close parentheses end cell row blank equals cell 3 times open parentheses negative integral fraction numerator 1 over denominator negative x squared plus 1 end fraction d x close parentheses end cell row blank equals cell 3 times open parentheses negative open parentheses fraction numerator ln space open vertical bar x plus 1 close vertical bar over denominator 2 end fraction close parentheses minus fraction numerator ln space open vertical bar x minus 1 close vertical bar over denominator 2 end fraction close parentheses plus C end cell row cell integral fraction numerator 3 over denominator x squared minus 1 end fraction d x end cell equals cell negative 3 open parentheses 1 half ln space open vertical bar x plus 1 close vertical bar minus 1 half ln space open vertical bar x minus 1 close vertical bar close parentheses plus C end cell end table 

Dengan demikian, nilai dari integral fraction numerator 3 over denominator x squared minus 1 end fraction d x adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 1 half ln space open vertical bar x plus 1 close vertical bar minus 1 half ln space open vertical bar x minus 1 close vertical bar close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank. end table 

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Pembahasan Soal:

Diperoleh intergalnya yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator d x over denominator 9 x squared plus 16 end fraction end cell equals cell integral fraction numerator d x over denominator 9 open parentheses x squared plus begin display style 16 over 9 end style close parentheses end fraction end cell row blank equals cell 1 over 9 integral fraction numerator 1 over denominator x squared plus begin display style 16 over 9 end style end fraction d x end cell end table

Ingatlah bahwa:

integral fraction numerator 1 over denominator x squared plus a squared end fraction d x equals 1 over a times arctan times x over a

Kemudian diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 over 9 integral fraction numerator 1 over denominator x squared plus begin display style 16 over 9 end style end fraction d x end cell equals cell 1 over 9 integral fraction numerator 1 over denominator x squared plus begin display style open parentheses 4 over 3 close parentheses squared end style end fraction d x end cell row blank equals cell 1 over 9 times fraction numerator 1 over denominator begin display style 4 over 3 end style end fraction times arctan times fraction numerator straight x over denominator begin display style 4 over 3 end style end fraction plus straight C end cell row blank equals cell 1 over 9 times 3 over 4 times arctan times straight x times 3 over 4 plus straight C end cell row blank equals cell 3 over 36 arctan fraction numerator 3 straight x over denominator 4 end fraction plus straight C end cell row blank equals cell fraction numerator arctan begin display style fraction numerator 3 straight x over denominator 4 end fraction end style over denominator 12 end fraction plus straight C end cell end table

Maka, hasil integral dari integral fraction numerator d x over denominator 9 x squared plus 16 end fraction adalah fraction numerator arctan begin display style fraction numerator 3 x over denominator 4 end fraction end style over denominator 12 end fraction plus C.

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Pembahasan Soal:

Misal: t equals sin to the power of negative 1 end exponent space x, maka:

table attributes columnalign right center left columnspacing 0px end attributes row t equals cell sin to the power of negative 1 end exponent space x end cell row cell d t end cell equals cell fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction d x end cell row cell d x end cell equals cell square root of 1 minus x squared end root space d t end cell end table

Diperoleh integralnya yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript 0 superscript 1 half end superscript fraction numerator sin to the power of negative 1 end exponent space x over denominator square root of 1 minus x squared end root end fraction d x end cell equals cell integral subscript 0 superscript 1 half end superscript sin to the power of negative 1 end exponent space x times fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction d x end cell row blank equals cell integral subscript 0 superscript 1 half end superscript space t space d t end cell row blank equals cell right enclose t squared over 2 end enclose subscript space 0 end subscript superscript space 1 half end superscript end cell row blank equals cell right enclose open parentheses sin to the power of negative 1 end exponent space x close parentheses squared over 2 end enclose subscript space 0 end subscript superscript space 1 half end superscript end cell row blank equals cell fraction numerator sin to the power of negative 1 end exponent open parentheses begin display style 1 half end style close parentheses squared over denominator 2 end fraction minus fraction numerator sin to the power of negative 1 end exponent open parentheses 0 close parentheses squared over denominator 2 end fraction end cell row blank equals cell fraction numerator begin display style straight pi squared over 36 end style over denominator 2 end fraction minus 0 end cell row blank equals cell straight pi squared over 36 times 1 half end cell row blank equals cell straight pi squared over 72 end cell end table

Maka, hasil integral dari integral subscript 0 superscript 1 half end superscript fraction numerator sin to the power of negative 1 end exponent space x over denominator square root of 1 minus x squared end root end fraction d x adalah straight pi squared over 72.

 

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