8 Contoh soal integral trigonometri dan pembahasan

Contoh soal integral trigonometri nomor 1

Nilai dari $\int_{0}^{\pi} cos \ x \ dx$ adalah …
A. 0
B. $\frac {1} {4}$
C. 2
D. 1
E. $\frac {1} {2}$

Pembahasan

Aturan integral trigonometri:

∫ cos x dx = sin x + C
∫ sin x dx = – cos x + C
∫ $\frac {1} {cos^{2} \ x}$ dx = tan x + C

Dengan menggunakan aturan di atas diperoleh hasil sebagai berikut.

$\int_{0}^{\pi} cos \ x \ dx$ = $\left [ sin \ x \right ]_{0}^{\pi}$
= (sin π – sin 0)
= (0 – 0) = 0

Soal ini jawabannya A.

Contoh soal integral trigonometri nomor 2

Nilai dari $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (2 \ sin \ x + cos \ x) \ dx$ adalah …
A. -1 – $\frac {1} {2} \sqrt {2}$
B. 1 + $\frac {1} {2} \sqrt {2}$
C. -2 + $\frac {1} {2} \sqrt {2}$
D. 2 + $\frac {1} {2} \sqrt {2}$
E. 2 – $\frac {1} {2} \sqrt {2}$

Pembahasan

$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (2 \ sin \ x + cos \ x) \ dx$ = $\left [ - 2 \ cos \ x + sin \ x \right ]_{\frac {\pi} {4}}^{\frac {\pi} {2}}$
= (- 2 cos $\frac {\pi} {2}$ + sin $\frac {\pi} {2}$ ) – (- 2 cos $\frac {\pi} {4}$ + sin $\frac {\pi} {4}$ )
= (- 2 . 0 + 1) – (- 2 . $\frac {1} {2} \sqrt {2}$ + $\frac {1} {2} \sqrt {2}$ )
= 1 – (- $\frac {1} {2} \sqrt {2}$ ) = 1 + $\frac {1} {2} \sqrt {2}$

Soal ini jawabannya B.

Contoh soal integral trigonometri nomor 3

Nilai dari $\int_{\frac{\pi}{2}}^{\pi} (2 \ sin \ x - 4 \ cos \ 2x) \ dx$ adalah …
A. -4
B. -2
C. 0
D. 2
E. 4

Pembahasan

$\int_{\frac{\pi}{2}}^{\pi} (2 \ sin \ x - 4 \ cos \ 2x) \ dx$ = $\left [ - 2 \ cos \ x - 4 . \frac {1} {2} . sin \ 2x \right ]_{\frac {\pi} {2}}^{\pi}$
= $\left [ - 2 \ cos \ x - 2 . sin \ 2x \right ]_{\frac {\pi} {2}}^{\pi}$
= (- 2 cos π – 2 sin 2π) – (-2 cos $\frac {\pi} {2}$ – 2 sin (2 . $\frac {\pi} {2}$ )
= ((- 2 . -1) – 2 . 0) – ((-2 . 0) – 2 . 0) = 2 – 0 – 0 = 2

Soal ini jawabannya D.

Contoh soal integral trigonometri nomor 4

Nilai dari ∫ x sin x dx adalah …
A. x cos x + C
B. -x cos x + C
C. x cos x + sin x + C
D. -x cos x + sin x + C
E. -x cos x – sin x + c

Pembahasan

Misalkan:

u = x maka du = dx
dv = sin x maka v = ∫ sin x dx = – cos x + C
∫u dv = uv – ∫v du
∫x sin x = -x cos x + c – ∫(- cos x) dx
∫x sin x = -x cos x + sin x + C

Soal ini jawabannya D.

Contoh soal integral trigonometri nomor 5

Hasil ∫ (cos x sin 3x) dx adalah …
A. – $\frac {1} {2}$ cos 4x – cos 2x + C
B. – $\frac {1} {4}$ cos 4x – $\frac {1} {2}$ cos 2x + C
C. – $\frac {1} {8}$ cos 4x – $\frac {1} {4}$ cos 2x + C
D. $\frac {1} {8}$ cos 4x + $\frac {1} {4}$ cos 2x + C
E. $\frac {1} {4}$ cos 4x + $\frac {1} {2}$ cos 2x + C

Pembahasan

sin 3x cos x = $\frac {1} {2}$ sin (3x + x) + $\frac {1} {2}$ sin (3x – x)
= $\frac {1} {2}$ sin 4x + $\frac {1} {2}$ sin 2x = $\frac {1} {2}$ (sin 4x + sin 2x)
∫ (cos x sin 3x) dx = ∫( $\frac {1} {2}$ (sin 4x + sin 2x) dx = $\frac {1} {2}$ ∫(sin 4x + sin 2x) dx
= $\frac {1} {2}$ (- $\frac {1} {4}$ cos 4x + (- $\frac {1} {2}$ cos 2x) + C
= – $\frac {1} {8}$ cos 4x – $\frac {1} {4}$ cos 2x + C

Soal ini jawabannya C.

Contoh soal integral trigonometri nomor 6

Hasil dari ∫ (sin⁵ 2x cos 2x) dx adalah …
A. – $\frac {1} {5}$ sin⁶ 2x + C
B. – $\frac {1} {10}$ sin⁶ 2x + C
C. – $\frac {1} {12}$ sin⁶ 2x + C
D. $\frac {1} {12}$ sin⁶ 2x + C
E. $\frac {1} {10}$ sin⁶ 2x + C

Pembahasan

Misalkan:

u = sin 2x maka du = 2 cos 2x dx atau dx = $\frac {du} {2 \ cos \ 2x}$
∫ (sin⁵ 2x cos 2x) dx = ∫u⁵ cos 2x $\frac {du} {2 \ cos \ 2x}$
= ∫ $\frac {1} {2}$ u⁵ du = $\frac {1} {2}$ ∫u⁵ du
= $\frac {1} {2}$ ( $\frac {1} {6}$ ) u⁶ + C
= $\frac {1} {12}$ sin⁶ 2x + C

Soal ini jawabannya D.

Contoh soal integral trigonometri nomor 7

Hasil dari ∫ (cos³ 2x sin 2x) dx = …
A. $\frac {1} {4}$ cos⁴ 2x + C
B. $\frac {1} {4}$ sin⁴ 2x + C
C. $\frac {1} {6}$ cos⁴ 2x + C
D. – $\frac {1} {8}$ cos⁴ 2x + C
E. – $\frac {1} {8}$ sin⁴ 2x + C

Pembahasan

u = cos 2x maka du = – 2 sin 2x dx atau dx = $\frac {du} {-2 \ sin \ 2x}$
∫ (cos³ 2x sin 2x) dx = ∫u³ sin 2x $\frac {du} {-2 \ sin \ 2x}$
= ∫- $\frac {1} {2}$ u³ du = – $\frac {1} {2}$ ∫u³ du
= – $\frac {1} {2}$ ( $\frac {1} {4}$ ) u⁴ + C
= – $\frac {1} {8}$ cos⁴ 2x + C

Soal ini jawabannya D.

Contoh soal integral trigonometri nomor 8

Hasil $\int_{0}^{\frac {\pi} {6}} (sin \ 4x \ cos \ 2x) \ dx$ = …
A. $\frac {4} {3}$
B. $\frac {2} {3}$
C. $\frac {1} {3}$
D. $\frac {7} {24}$
E. – $\frac {1} {3}$

Pembahasan

sin 4x cos 2x = $\frac {1} {2}$ sin (4x + 2x) + $\frac {1} {2}$ sin (4x – 2x)
= $\frac {1} {2}$ sin 6x + $\frac {1} {2}$ sin 2x = $\frac {1} {2}$ (sin 6x + sin 2x)
$\int_{0}^{\frac {\pi} {6}} (sin \ 4x \ cos \ 2x) \ dx$ = $\frac {1} {2} \int_{0}^{\frac {\pi} {6}} (sin \ 6x \ + sin \ 2x) \ dx$
= $\frac {1} {2}$ ( $\left [ - \frac {1} {6} \ cos \ 6x - \frac {1} {2} cos \ 2x \right ]_{0}^{\frac {\pi} {6}}$ )
= – $\frac {1} {2}$ ( $\frac {1} {6}$ cos π + $\frac {1} {2}$ cos $\frac {\pi} {3}$ ) – ( $\frac {1} {6}$ cos 0 + $\frac {1} {2}$ cos 0))
= – $\frac {1} {2}$ (- $\frac {1} {6}$ + $\frac {1} {2}$ ) – ( $\frac {1} {6}$ + $\frac {1} {2}$ )
= – $\frac {1} {2}$ (- $\frac {1} {6}$ = $\frac {1} {12}$

Jawaban: –

8 Contoh soal integral trigonometri dan pembahasan

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