A-Level Edexcel Maths Pure Implicit Differentiation: 2. (a) Use integration by parts

2. (a) Use integration by parts to find $ \int xe^{x} \, dx $ - Edexcel - A-Level Maths Pure - Question 4 - 2008 - Paper 7

Question 4

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2. (a) Use integration by parts to find $ \int xe^{x} \, dx $. (b) Hence find $ \int x e^{2x} \, dx $.

Worked Solution & Example Answer:2. (a) Use integration by parts to find $ \int xe^{x} \, dx $ - Edexcel - A-Level Maths Pure - Question 4 - 2008 - Paper 7

Step 1

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Answer

We start by applying the integration by parts formula, which is:

$\int u \, dv = uv - \int v \, du$

Assign:
- Let ( u = x ) and ( dv = e^{x} , dx )
- This gives us ( du = dx ) and ( v = e^{x} )
Apply the formula: $\int xe^{x} \, dx = xe^{x} - \int e^{x} \, dx$
Calculate the integral of ( e^{x} ): $\int e^{x} \, dx = e^{x}$
Therefore, substituting back: $\int xe^{x} \, dx = xe^{x} - e^{x} + c$
This simplifies to: $\int xe^{x} \, dx = e^{x} (x - 1) + c$

Step 2

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Answer

Now, we use the result from part (a) to find the integral ( \int x e^{2x} , dx ). Again we apply integration by parts:

Assign:
- Let ( u = x ) and ( dv = e^{2x} , dx )
- So, ( du = dx ) and ( v = \frac{1}{2} e^{2x} )
Apply the formula: $\int xe^{2x} \, dx = x \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \, dx$
Calculate the integral of ( e^{2x} ): $\int e^{2x} \, dx = \frac{1}{2} e^{2x}$
Substituting back: $\int xe^{2x} \, dx = \frac{1}{2} x e^{2x} - \frac{1}{2} \cdot \frac{1}{2} e^{2x} + c$
Thus we have: $\int xe^{2x} \, dx = \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} + c$