Math Problem Statement

Solve it with parts

If ∫(2x - 1)e^(2x + 3) dx = yz - ∫z dy, then ∫z dy = ...

Solution

We are given the following equation to solve:

(2x1)e2x+3dx=yzzdy\int (2x - 1) e^{2x+3} \, dx = yz - \int z \, dy and the task is to find zdy\int z \, dy.

Step 1: Solving the left side (2x1)e2x+3dx\int (2x - 1) e^{2x+3} \, dx

To solve this, we will use integration by parts.

Let:

  • u=2x1u = 2x - 1, hence du=2dxdu = 2 \, dx.
  • dv=e2x+3dxdv = e^{2x+3} \, dx, hence v=12e2x+3v = \frac{1}{2} e^{2x+3} (since the derivative of 2x+32x + 3 is 2).

Now apply integration by parts: udv=uvvdu\int u \, dv = u v - \int v \, du Substituting the values: (2x1)e2x+3dx=(2x1)12e2x+312e2x+32dx\int (2x - 1) e^{2x+3} \, dx = (2x - 1) \cdot \frac{1}{2} e^{2x+3} - \int \frac{1}{2} e^{2x+3} \cdot 2 \, dx This simplifies to: 12(2x1)e2x+3e2x+3dx\frac{1}{2} (2x - 1) e^{2x+3} - \int e^{2x+3} \, dx The integral of e2x+3e^{2x+3} is: 12e2x+3\frac{1}{2} e^{2x+3} So, the result becomes: 12(2x1)e2x+312e2x+3\frac{1}{2} (2x - 1) e^{2x+3} - \frac{1}{2} e^{2x+3} Simplifying further: 12e2x+3[(2x1)1]=12e2x+3(2x2)=e2x+3(x1)\frac{1}{2} e^{2x+3} [(2x - 1) - 1] = \frac{1}{2} e^{2x+3} (2x - 2) = e^{2x+3} (x - 1)

Step 2: Rearranging the equation

We are given that: (2x1)e2x+3dx=yzzdy\int (2x - 1) e^{2x+3} \, dx = yz - \int z \, dy From step 1, we know: e2x+3(x1)=yzzdye^{2x+3} (x - 1) = yz - \int z \, dy Rearranging to find zdy\int z \, dy: zdy=yze2x+3(x1)\int z \, dy = yz - e^{2x+3} (x - 1)

Step 3: Comparing with the given options

Looking at the structure, the closest match to zdy\int z \, dy from the options appears to be:

12e2x+3+c\boxed{\frac{1}{2} e^{2x+3} + c}

which matches option (b).


Would you like further clarification on any steps or have additional questions? Here are some questions you might find helpful:

  1. What is integration by parts, and when is it used?
  2. How do we determine the choice of uu and dvdv in integration by parts?
  3. What is the integral of an exponential function like e2x+3e^{2x+3}?
  4. How can you verify the solution to a definite integral?
  5. What are the conditions for applying the method of substitution in integrals?

Tip: For solving complex integrals, it's important to identify which method (substitution, integration by parts, etc.) will simplify the expression the most effectively.

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Math Problem Analysis

Mathematical Concepts

Integration by Parts
Exponential Functions

Formulas

∫u dv = uv - ∫v du (Integration by Parts)
Integral of e^(ax+b) is (1/a) * e^(ax+b)

Theorems

Integration by Parts

Suitable Grade Level

University Level - Calculus