Question

Find ∫(cos√x) dx=?

Answer 1

Let's solve the integral \(\int \cos(\sqrt{x}) \, dx\).

1. Substitution: Let \( u = \sqrt{x} \). Then \( x = u^2 \), and \( dx = 2u \, du \). Substituting these into the integral, we have: \[ \int \cos(\sqrt{x}) \, dx = \int \cos(u) \cdot 2u \, du = 2 \int u \cos(u) \, du \]

2. Integration by Parts: Now, we need to use integration by parts to solve \(\int u \cos(u) \, du\). 
Recall the integration by parts formula: \[ \int v \, dw = vw - \int w \, dv \] Let \( v = u \) and \( dw = \cos(u) \, du \). Then \( dv = du \) and \( w = \sin(u) \). Applying the formula: \[ \int u \cos(u) \, du = u \sin(u) - \int \sin(u) \, du \] \[ \int u \cos(u) \, du = u \sin(u) - (-\cos(u)) + C_1 \] \[ \int u \cos(u) \, du = u \sin(u) + \cos(u) + C_1 \]

3. Substitute Back: Now, substitute this result back into our integral and then substitute \( u = \sqrt{x} \): \[ 2 \int u \cos(u) \, du = 2(u \sin(u) + \cos(u)) + C \] \[ 2 \int u \cos(u) \, du = 2(\sqrt{x} \sin(\sqrt{x}) + \cos(\sqrt{x})) + C \]

Final Answer: Therefore, the integral is: \[ \int \cos(\sqrt{x}) \, dx = 2\sqrt{x} \sin(\sqrt{x}) + 2\cos(\sqrt{x}) + C \]