Question

in exercises 21 - 24, solve the initial value problem using the fundamental theorem. (your answer will contain a definite integral.)

1. \\( \frac{d y}{d x}=\sin \left(x^{2}\

ight) \\) and \\( y = 5 \\) when \\( x = 1 \\)

Explanation:

Step1: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if \(\frac{dy}{dx}=f(x)\), then \(y = y(a)+\int_{a}^{x}f(t)dt\). Here \(f(x)=\sin(x^{2})\), \(a = 1\) and \(y(1)=5\).

Step2: Write the solution

Substitute the values into the formula: \(y=5+\int_{1}^{x}\sin(t^{2})dt\)

Answer:

\(y = 5+\int_{1}^{x}\sin(t^{2})dt\)