Pembahasan Latihan Soal Integral (2) UN SMA

$\int$ x sin 2x dx = …

A. $\frac{1}{4}$ sin 2x – $\frac{1}{2}$ x cos 2x + C

B. $\frac{1}{4}$ sin 2x + $\frac{1}{2}$ x cos 2x + C

C. $\frac{1}{4}$ sin 2x – $\frac{1}{2}$ x cos 2x + C

D. - $\frac{1}{4}$ cos 2x – $\frac{1}{2}$ x sin 2x + C

E. $\frac{1}{4}$ cos 2x + $\frac{1}{2}$ x sin 2x + C

PEMBAHASAN :

misal u = x $\Rightarrow$ du = dx

dv = sin 2x dx $\Rightarrow$ v = - $\frac{1}{2}$ cos 2x

$\int$ u dv = uv – $\int$ v du

= - $\frac{1}{2}$ x cos 2x – $\int$ - $\frac{1}{2}$ cos 2x dx

= - $\frac{1}{2}$ x cos 2x + $\frac{1}{4}$ sin 2x + C

= $\frac{1}{4}$ sin 2x – $\frac{1}{2}$ x cos 2x + C

JAWABAN : C
$\int_0^{\frac{\pi}{2}}$ (sin² x – cos² x) dx = …

A. - $\frac{1}{2}$

B. - $\frac{1}{2}$ $\pi$

C. 0

D. $\frac{1}{2}$

E. $\frac{1}{2}$ $\pi$

PEMBAHASAN :

$\int_0^{\frac{\pi}{2}}$ (sin² x – cos² x) dx = $\int_0^{\frac{\pi}{2}}$ cos 2x dx

[ingat Sifat Dasar Trigonometri]

misal u = 2x $\Rightarrow$ du = 2 dx

= $\int_0^{\frac{\pi}{2}}$ cos u $\frac{du}{2}$

= $\frac{1}{2}$ sin u $\mid_0^{\frac{\pi}{2}}$

Substitusi kembali u = 2x

= $\frac{1}{2}$ sin 2x $\mid_0^{\frac{\pi}{2}}$

= $\frac{1}{2}$ [sin 2 $\frac{\pi}{2}$ – sin 2.0]

= 0

JAWABAN : C
Hasil $\int$ 2x cos $\frac{1}{2}$ x dx = …

A. 4x sin $\frac{1}{2}$ x + 8 cos $\frac{1}{2}$ x + C

B. 4x sin $\frac{1}{2}$ x – 8 cos $\frac{1}{2}$ x + C

C. 4x sin $\frac{1}{2}$ x + 4 cos $\frac{1}{2}$ x + C

D. 4x sin $\frac{1}{2}$ x – 8 cos $\frac{1}{2}$ x + C

E. 4x sin $\frac{1}{2}$ x + 2 cos $\frac{1}{2}$ x + C

PEMBAHASAN :

disini akan digunakan Integral Parsial

misal u = 2x $\Rightarrow$ du = 2 dx

dv = cos $\frac{1}{2}$ x $\Rightarrow$ v = 2 sin $\frac{1}{2}$ x

$\int$ 2x cos $\frac{1}{2}$ x dx = 2x 2 sin $\frac{1}{2}$ x – $\int$ 2 sin $\frac{1}{2}$ x 2 dx

= 4x sin $\frac{1}{2}$ x – 4 (-2 cos $\frac{1}{2}$ x) + C

= 4x sin $\frac{1}{2}$ x + 8 cos $\frac{1}{2}$ x + C

JAWABAN : A
Hasil $\int$ $x\sqrt{9-x^2}$ dx = …

A. - $\frac{1}{3}$ (9 – x²) $x\sqrt{9-x^2}$ + C

B. - $\frac{2}{3}$ (9 – x²) $x\sqrt{9-x^2}$ + C

C. $\frac{2}{3}$ (9 – x²) $x\sqrt{9-x^2}$ + C

D. $\frac{2}{3}$ (9 – x²) $x\sqrt{9-x^2}$ + $\frac{2}{9}$ (9 – x²) $x\sqrt{9-x^2}$ + C

E. $\frac{1}{3}$ (9 – x²) $x\sqrt{9-x^2}$ + $\frac{1}{9}$ (9 – x²) $x\sqrt{9-x^2}$ + C

PEMBAHASAN :

misal u = 9 – x² $\Rightarrow$ du = -2x dx

= $\int$ $\sqrt{u}$ $\frac{du}{-2}$

= $\int$ $-\frac{1}{2}$ u^1/2 du

= $-\frac{1}{2}$ . $\frac{2}{3}$ u^3/2 + C

substitusi u = 9 – x², sehingga diperoleh

= $-\frac{1}{3}$ (9 – x²)^3/2 + C

= $-\frac{1}{3}$ (9 – x²) $\sqrt{9-x^2}$ + C

JAWABAN :
Nilai $\int_0^1$ 5x(1 – x)⁶ dx = …

A. 75/56

B. 10/56

C. 5/56

D. -7/56

E. -10/56

PEMBAHASAN :

misal u = 5x $\Rightarrow$ du = 5 dx

dv = (1 – x)⁶ dx $\Rightarrow$ v = $-\frac{1}{7}$ (1 – x)⁷

$\int_0^1$ 5x(1 – x)⁶ dx = 5x $-\frac{1}{7}$ (1 – x) – $\int_0^1$ $-\frac{1}{7}$ (1 – x)⁷ 5 dx

= $-\frac{5}{7}$ x(1 – x) + $\int_0^1$ $\frac{1}{7}$ . $\frac{1}{8}$ (1 – x)⁸ 5 dx

= ( $-\frac{5}{7}$ x(1 – x)⁷ + $\frac{5}{56}$ (1 – x)⁸) $\mid_0^1$

= ( $-\frac{5}{7}$ .1.(1 – 1)⁷ + $\frac{5}{56}$ (1 – 1)⁸) – ( $-\frac{5}{7}$ .0.(1 – 0)⁷ + $\frac{5}{56}$ (1 – 0)⁸)

= (0 + 0) – (0 + $\frac{5}{56}$ )

= $\frac{5}{56}$

JAWABAN : C
Hasil dari $\int$ cos x cos 4x dx = …

A. - $\frac{1}{5}$ sin 5x – $\frac{1}{3}$ x sin 3x + C

B. $\frac{1}{10}$ sin 5x + $\frac{1}{6}$ x sin 3x + C

C. $\frac{2}{5}$ sin 5x + $\frac{2}{3}$ x sin 3x + C

D. $\frac{1}{2}$ cos 5x + $\frac{1}{2}$ x cos 3x + C

E. - $\frac{1}{2}$ sin 5x – $\frac{1}{2}$ x sin 3x + C

PEMBAHASAN :

$\int$ cos x cos 4x dx = $\int$ $\frac{1}{2}$ (cos 5x + cos 3x) dx

= $\int$ $\frac{1}{2}$ cos 5x dx + $\int$ $\frac{1}{2}$ cos 3x dx

misal u = 5x $\Rightarrow$ du = 5 dx

v = 3x $\Rightarrow$ dv = 3 dx

substitusi, sehingga

= $\int$ $\frac{1}{2}$ cos u $\frac{du}{5}$ + $\int$ $\frac{1}{2}$ cos v $\frac{dv}{3}$

= $\frac{1}{10}$ sin u + $\frac{1}{6}$ sin v + C

substitusi kembali u = 5x dan v = 3x

= $\frac{1}{10}$ sin 5x + $\frac{1}{6}$ sin 3x + C

JAWABAN : B
Hasil dari $\int$ cos⁴ 2x sin 2x dx = …

A. $-\frac{1}{10}$ sin⁵ 2x + C

B. $-\frac{1}{10}$ cos⁵ 2x + C

C. $-\frac{1}{5}$ cos⁵ 2x + C

D. $\frac{1}{5}$ cos⁵ 2x + C

E. $\frac{1}{10}$ sin⁵ 2x + C

PEMBAHASAN :

misal u = cos 2x $\Rightarrow$ du = -2 sin 2x dx

$\int$ cos⁴ 2x sin 2x dx = $\int$ u⁴ $-\frac{du}{2}$

= $-\frac{1}{2}$ $\frac{1}{5}$ u⁵ + C

substitusi kembali u = cos 2x

= $-\frac{1}{10}$ cos⁵ 2x + C

JAWABAN : B
Hasil dari $\int$ 4 sin 5x cos 3x dx = …

A. -2 cos 8x – 2 cos 2x + C

B. $-\frac{1}{4}$ cos 8x – 2 cos 2x + C

C. $\frac{1}{4}$ cos 8x + 2 cos 2x + C

D. $-\frac{1}{2}$ cos 8x – 2 cos 2x + C

E. $\frac{1}{2}$ cos 8x – 2 cos 2x + C

PEMBAHASAN :

$\int$ 4 sin 5x cos 3x dx = $\int$ 2(sin 8x + sin 2x) dx

= $\int$ 2(sin 8x + sin 2x) dx

= $\int$ 2 sin 8x dx + $\int$ 2 sin 2x dx

misal u = 8x $\Rightarrow$ du = 8

v = 2x $\Rightarrow$ du = 2

substitusi, sehingga

= $\int$ 2 sin u $\frac{du}{8}$ + $\int$ 2 sin v $\frac{dv}{2}$

= $\frac{1}{4}$ $\int$ sin u du + $\int$ sin v dv

= $-\frac{1}{4}$ cos u – cos v + C

substitusi kembali u = 8x dan v = 2x

= $-\frac{1}{4}$ cos 8x – cos 2x + C

JAWABAN : B
Hasil dari $\int$ x² sin 2x dx = …

A. $-\frac{1}{2}$ x² cos 2x – $\frac{1}{2}$ x sin 2x + $\frac{1}{4}$ cos 2x + C

B. $-\frac{1}{2}$ x² cos 2x + $\frac{1}{2}$ x sin 2x – $\frac{1}{4}$ cos 2x + C

C. $-\frac{1}{2}$ x² cos 2x + $\frac{1}{2}$ x sin 2x + $\frac{1}{4}$ cos 2x + C

D. $\frac{1}{2}$ x² cos 2x – $\frac{1}{2}$ x sin 2x – $\frac{1}{4}$ cos 2x + C

E. $\frac{1}{2}$ x² cos 2x – $\frac{1}{2}$ x sin 2x + $\frac{1}{4}$ cos 2x + C

PEMBAHASAN :

disini kita akan menggunakan Integral Parsial

misal u = x² $\Rightarrow$ du = 2x dx

dv = sin 2x dx $\Rightarrow$ v = $-\frac{1}{2}$ cos 2x

$\int$ x² sin 2x dx = (x²) $-\frac{1}{2}$ cos 2x – $\int$ $-\frac{1}{2}$ cos 2x 2x dx

= $-\frac{1}{2}$ x² cos 2x + $\int$ x cos 2x dx

Integral Parsial lagi

misal u = x $\Rightarrow$ du = dx

dv = cos 2x dx $\Rightarrow$ v = $\frac{1}{2}$ sin 2x

= $-\frac{1}{2}$ x² cos 2x + [x $\frac{1}{2}$ sin 2x – $\int$ $\frac{1}{2}$ sin 2x dx]

= $-\frac{1}{2}$ x² cos 2x + $\frac{1}{2}$ x sin 2x + $\frac{1}{4}$ cos 2x

JAWABAN : C
Hasil dari $\int$ sin² x cos x dx = …

A. $\frac{1}{3}$ cos³ x + C

B. $-\frac{1}{3}$ cos³ x + C

C. $-\frac{1}{3}$ sin³ x + C

D. $\frac{1}{3}$ sin³ x + C

E. 3 sin³ x + C

PEMBAHASAN :

misal u = sin x $\Rightarrow$ du = cos x du,

kemudian substitusi, sehingga

$\int$ sin² x cos x dx = $\int$ u² du

= $\frac{1}{3}$ u³ + C

substitusi kembali u = sin x,

= $\frac{1}{3}$ sin³ x + C

JAWABAN : D

NOTE : silahkan dikoreksi dan berikan komentar jika ada kesalahan atau masih ada keambiguan dalam penyelesaian soal-soal ini.

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