If f is continuous and int _(0)^4f(x)dx=10 find int _(0)^2f(2x)dx

To find the value of \(\int_{0}^{2} f(2x) \, dx\), we can use a substitution method. Let's set \(u = 2x\). Then, \(du = 2 \, dx\), which implies \(dx = \frac{1}{2} \, du\).<br /><br />Now, we need to change the limits of integration accordingly. When \(x = 0\), \(u = 2 \cdot 0 = 0\). When \(x = 2\), \(u = 2 \cdot 2 = 4\).<br /><br />Substituting these into the integral, we get:<br /><br />\[<br />\int_{0}^{2} f(2x) \, dx = \int_{0}^{4} f(u) \cdot \frac{1}{2} \, du<br />\]<br /><br />Since \(\int_{0}^{4} f(x) \, dx = 10\), we can write:<br /><br />\[<br />\int_{0}^{4} f(u) \cdot \frac{1}{2} \, du = \frac{1}{2} \int_{0}^{4} f(u) \, du = \frac{1}{2} \cdot 10 = 5<br />\]<br /><br />Therefore, the value of \(\int_{0}^{2} f(2x) \, dx\) is \(5\).