Jika \( \displaystyle \int_0^4 f(x) \ dx = \sqrt{2} \) maka nilai \( \displaystyle \int_0^2 x f(x^2) \ dx \) adalah…
- \( \frac{\sqrt{2}}{4} \)
- \( \frac{\sqrt{2}}{2} \)
- \( \sqrt{2} \)
- \( 2\sqrt{2} \)
- \( 4\sqrt{2} \)
(SBMPTN 2018)
Pembahasan:
Untuk mengerjakan soal ini, kita perlu mengingat ini:
\begin{aligned} \int_a^b \ f(x) \ dx = \int_a^b f(u) \ du = \int_a^b f(t) \ dt = \cdots \end{aligned}
Sekarang, misalkan \(u = x^2 \) sehingga diperoleh:
\begin{aligned} u = x^2 \Leftrightarrow \frac{du}{dx} &= 2x \\[8pt] dx &= \frac{du}{2x} \end{aligned}
Selanjutnya, ganti batas dari integralnya, yakni:
\begin{aligned} x = 0 \Rightarrow u = x^2 = 0^2 = 0 \\[8pt] x = 2 \Rightarrow u = x^2 = 2^2 = 4 \end{aligned}
Berdasarkan hasil di atas, maka diperoleh berikut ini:
\begin{aligned} \int_0^2 x f(x^2) \ dx &= \int_0^4 x f(u) \cdot \frac{du}{2x} \\[8pt] &= \frac{1}{2} \int_0^4 f(u) \ du \\[8pt] &= \frac{1}{2} \int_0^4 f(x) \ dx \\[8pt] &= \frac{1}{2}\sqrt{2} \end{aligned}
Jawaban B.