Question

In Exercises \(21-26,\) construct a function of the form \(y=\int^{x} f(t) d t+C\) that satisfies the given conditions. $$\frac{d y}{d x}=\cos ^{2} 5 x,\( and \)y=-2\( when \)x=7$$

Short answer

So the function is \(y= \frac{1}{2} x + \frac{1}{20} \sin 10x - 2 - \frac{7}{2} - \frac{\sin 70}{20}\).

Step-by-step solution

Integrate the Function

The anti-derivative of the integral \(\cos^2 5x\) can be found using the power-reduction identity \(\cos^2x = (1 + \cos 2x)/2\). This leads to:\[y=\int^{x} \cos^{2} 5t dt + C = \int^{x} \left(\frac{1}{2} + \frac{1}{2} \cos 2(5t)\right) dt + C\]

Simplify the Integral

Separate and integrate term by term:\[y= \int^{x} \frac{1}{2} dt + \int^{x} \frac{1}{2} \cos 10t dt + C\]This simplifies to:\[y= \frac{1}{2} x + \frac{1}{20} \sin 10x + C\]

Find constant of Integration

Substitute the given coordinates, \(x=7\), \(y=-2\), into the equation from step 2:\[-2= \frac{1}{2} * 7 + \frac{1}{20} \sin 70 + C\]Solve for \(C\) to get:\[C= -2 - \frac{7}{2} - \frac{\sin 70}{20}\]