Differentiate 3xcosx and HENCE integrate 2x * sin x dx from 0 to pi/2

Let's go through this step-by-step.

### Step 1: Differentiate \( 3x \cos x \)
To differentiate the function \( 3x \cos x \), we will apply the product rule. The product rule states:

\[
\frac{d}{dx}(uv) = u'v + uv'
\]

Here, let:
- \( u = 3x \), so \( u' = 3 \)
- \( v = \cos x \), so \( v' = -\sin x \)

Now, applying the product rule:

\[
\frac{d}{dx}(3x \cos x) = 3 \cdot \cos x + 3x \cdot (-\sin x)
\]

Simplifying:

\[
\frac{d}{dx}(3x \cos x) = 3 \cos x - 3x \sin x
\]

### Step 2: HENCE integrate \( \int_0^{\frac{\pi}{2}} 2x \sin x \, dx \)

We notice that the form of \( 2x \sin x \) is very similar to the derivative of \( 3x \cos x \). From Step 1, we found:

\[
\frac{d}{dx}(3x \cos x) = 3 \cos x - 3x \sin x
\]

We can rearrange this to find \( x \sin x \):

\[
x \sin x = \frac{1}{3}(3 \cos x - \frac{d}{dx}(3x \cos x))
\]

Multiplying both sides by 2 gives us:

\[
2x \sin x = \frac{2}{3}(3 \cos x) - \frac{2}{3} \frac{d}{dx}(3x \cos x)
\]

Now, we can integrate \( 2x \sin x \) from 0 to \( \frac{\pi}{2} \):

\[
\int_0^{\frac{\pi}{2}} 2x \sin x \, dx = \frac{2}{3} \int_0^{\frac{\pi}{2}} 3 \cos x \, dx - \frac{2}{3} \left[ 3x \cos x \right]_0^{\frac{\pi}{2}}
\]

Now, let's solve these integrals:

1. First, integrate \( \cos x \):

\[
\int_0^{\frac{\pi}{2}} 3 \cos x \, dx = 3 [\sin x]_0^{\frac{\pi}{2}} = 3 [1 - 0] = 3
\]

2. Then, evaluate \( 3x \cos x \) at \( x = 0 \) and \( x = \frac{\pi}{2} \):

\[
3x \cos x \Big|_0^{\frac{\pi}{2}} = 3 \cdot \frac{\pi}{2} \cdot \cos \left( \frac{\pi}{2} \right) - 3 \cdot 0 \cdot \cos(0) = 0 - 0 = 0
\]

Thus, the integral becomes:

\[
\int_0^{\frac{\pi}{2}} 2x \sin x \, dx = \frac{2}{3} \cdot 3 - 0 = 2
\]

### Final Answer:
The value of \( \int_0^{\frac{\pi}{2}} 2x \sin x \, dx \) is \( \boxed{2} \).

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Do you want further details on any of the steps, or do you have questions?  
Here are some related questions for practice:

1. Differentiate \( x^2 \sin x \).
2. Integrate \( \int x \cos x \, dx \).
3. Find the integral of \( \int_0^{\pi} x \sin x \, dx \).
4. Differentiate \( x \tan x \).
5. Solve \( \int_0^{\pi/2} \sin^2 x \, dx \).

**Tip:** The product rule is key in problems involving functions multiplied together, and it's often paired with the chain rule for complex functions.

Learn how to differentiate the function 3xcosx using the product rule and use the result to integrate 2x * sin x from 0 to pi/2. This problem involves differentiation and definite integration techniques suitable for high school students in Grades 11-12.

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