Integration using u-substitution

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**Step 1: Choose a substitution.**  
Identify a part of the integrand whose derivative also appears. For example, consider  
$$
\int 2x\cos(x^2)\,dx.
$$  
Let  
$$
u = x^2.
$$

**Step 2: Compute $$ du $$.**  
Differentiate $$ u $$ with respect to $$ x $$:
$$
\frac{du}{dx} = 2x \quad \Longrightarrow \quad du = 2x\,dx.
$$

**Step 3: Rewrite the integral in terms of $$ u $$.**  
Since $$ du = 2x\,dx $$, the integral becomes:
$$
\int \cos(x^2) \cdot 2x\,dx = \int \cos(u)\,du.
$$

**Step 4: Integrate with respect to $$ u $$.**  
The integral of $$ \cos(u) $$ is:
$$
\int \cos(u)\,du = \sin(u) + C.
$$

**Step 5: Substitute back to the original variable.**  
Recall that $$ u = x^2 $$. Thus, the final result is:
$$
\sin(x^2) + C.
$$

This demonstrates the technique of u-substitution to solve the integral.
To integrate $$ 2x\cos(x^2)\,dx $$ using u-substitution: 1. Let $$ u = x^2 $$. 2. Then, $$ du = 2x\,dx $$. 3. Substitute into the integral: $$ \int \cos(u)\,du = \sin(u) + C $$. 4. Substitute back: $$ \sin(x^2) + C $$. So, the integral equals $$ \sin(x^2) + C $$.

Integration using u-substitution

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