Exercise Zone : Integral Tak Tentu

Table of Contents

Berikut ini adalah kumpulan soal mengenai integral tak tentu tipe standar. Jika ingin bertanya soal, silahkan gabung ke grup Facebook atau Telegram.

Tipe:

StandarSNBTHOTS

No.

\displaystyle\int\dfrac{x+2}{\sqrt{x^2+4x-6}}\ dx= ....

1. \dfrac18\sqrt{x^2+4x-6}+C
2. \dfrac14\sqrt{x^2+4x-6}+C
3. \dfrac12\sqrt{x^2+4x-6}+C

4. \sqrt{x^2+4x-6}+C
5. 2\sqrt{x^2+4x-6}+C

Matematika Zone Id

ALTERNATIF PENYELESAIAN

Misal u = x² + 4x − 6 \begin{aligned} du&=(2x+4)\ dx\\ du&=2(x+2)\ dx\\ (x+2)\ dx&=\dfrac12du \end{aligned}

\begin{aligned} \displaystyle\int\dfrac{x+2}{\sqrt{x^2+4x-6}}\ dx&=\dfrac12\displaystyle\int\dfrac1{\sqrt{u}}\ du\\ &=\dfrac12\displaystyle\int\dfrac1{u^{\frac12}}\ du\\ &=\dfrac12\displaystyle\int u^{-\frac12}\ du\\ &=\dfrac12\left(2u^{\frac12}\right)+C\\ &=\sqrt{u}+C\\ &=\boxed{\boxed{\sqrt{x^2+4x-6}+C}} \end{aligned}

Jadi, \displaystyle\int\dfrac{x+2}{\sqrt{x^2+4x-6}}\ dx=\sqrt{x^2+4x-6}+C.
JAWAB: D

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No.

Hasil dari \displaystyle\int-720x^2\sqrt{9-4x^3}\ dx=

1. {80\left(9-4x^3\right)\sqrt{9-4x^3}+C}
2. {60\left(9-4x^3\right)\sqrt{9-4x^3}+C}
3. {40\left(9-4x^3\right)\sqrt{9-4x^3}+C}

4. {20\left(9-4x^3\right)\sqrt{9-4x^3}+C}
5. {10\left(9-4x^3\right)\sqrt{9-4x^3}+C}

Matematika Zone Id

ALTERNATIF PENYELESAIAN

Misal u = 9 − 4x³
du = −12x² dx \begin{aligned} \displaystyle\int-720x^2\sqrt{9-4x^3}\ dx&=\displaystyle\int60\left(-12x^2\right)\sqrt{9-4x^3}\ dx\\ &=\displaystyle\int60\sqrt{u}\ du\\ &=\displaystyle\int60u^{\frac12}\ du\\ &=60\cdot\dfrac23u^{\frac32}+C\\ &=40u\sqrt{u}+C\\ &=\boxed{\boxed{40\left(9-4x^3\right)\sqrt{9-4x^3}+C}} \end{aligned}

Jadi, \displaystyle\int-720x^2\sqrt{9-4x^3}\ dx=40\left(9-4x^3\right)\sqrt{9-4x^3}+C.
JAWAB: C

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No.

\displaystyle\int\left(\dfrac1{\sqrt{x}}-\sqrt{x}\right)^2\ dx adalah

1. \dfrac12x^2-2x+\ln x+C
2. x^2-2+\dfrac1x+C
3. x² − 2x + x⁻¹ + C

4. \dfrac12x^2-2x+x^{-1}+C
5. \dfrac12x^2-2+\dfrac1x+C

Matematika Zone Id

ALTERNATIF PENYELESAIAN

\begin{aligned} \displaystyle\int\left(\dfrac1{\sqrt{x}}-\sqrt{x}\right)^2\ dx&=\displaystyle\int\left(\left(\dfrac1{\sqrt{x}}\right)^2-2\cdot\dfrac1{\sqrt{x}}\cdot\sqrt{x}+\left(\sqrt{x}\right)^2\right)\ dx\\ &=\displaystyle\int\left(\dfrac1x-2+x\right)\ dx\\ &=\ln x-2x+\dfrac12x^2+C\\ &=\boxed{\boxed{\dfrac12x^2-2x+\ln x+C}} \end{aligned}

Jadi, \displaystyle\int\left(\dfrac1{\sqrt{x}}-\sqrt{x}\right)^2\ dx=\dfrac12x^2-2x+\ln x+C.
JAWAB: A

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No.

\displaystyle\int2\sqrt{x}\ dx= ...

1. \sqrt{x}+C
2. x\sqrt{x}+C

3. 2\sqrt{x}+C
4. \dfrac43x\sqrt{x}+C

Matematika Zone Id

ALTERNATIF PENYELESAIAN

\begin{aligned} \displaystyle\int2\sqrt{x}\ dx&=\displaystyle\int2x^{\frac12}\ dx\\ &=2\cdot\dfrac23x^{\frac32}+C\\ &=\boxed{\boxed{\dfrac43x\sqrt{x}+C}} \end{aligned}

Jadi, \displaystyle\int2\sqrt{x}\ dx=\dfrac43x\sqrt{x}+C.
JAWAB: D

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No.

\int\frac{e^{8+\frac1x}}{x^2}\ dx=....

1. \dfrac{e^{8+\frac1x}}{x^2}+C
2. -\dfrac{e^{8+\frac1x}}{x^2}+C

3. e^{8+\frac1x}+C
4. -e^{8+\frac1x}+C

Matematika Zone Id

ALTERNATIF PENYELESAIAN

Misalkan u=8+\dfrac1x.
du=-\dfrac1{x^2}\ dx \begin{aligned} \int\frac{e^{8+\frac1x}}{x^2}\ dx&=-\int\left(e^{8+\frac1x}\right)\left(-\frac1{x^2}\right)\ dx\\ &=-\int e^u\ du\\ &=-e^u+C\\ &=\boxed{\boxed{-e^{8+\frac1x}+C}} \end{aligned}

Jadi, \int\frac{e^{8+\frac1x}}{x^2}\ dx=-e^{8+\frac1x}+C.
JAWAB: D

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