6. $$ \int\left(1-e^{x}\right)^{2} d x $$

Question

UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ \int(1-e^{x})^{2} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \left(1-e^{x}\right)^{2} dx$$
- step1: Calculate:
  $$\int \left(1-2e^{x}+e^{2x}\right) dx$$
- step2: Use properties of integrals:
  $$\int 1 dx+\int -2e^{x} dx+\int e^{2x} dx$$
- step3: Evaluate the integral:
  $$x+\int -2e^{x} dx+\int e^{2x} dx$$
- step4: Evaluate the integral:
  $$x-2e^{x}+\int e^{2x} dx$$
- step5: Evaluate the integral:
  $$x-2e^{x}+\frac{e^{2x}}{2}$$
- step6: Add the constant of integral C:
  $$x-2e^{x}+\frac{e^{2x}}{2} + C, C \in \mathbb{R}$$
The integral of $$ (1-e^{x})^{2} $$ with respect to $$ x $$ is $$ x-2e^{x}+\frac{e^{2x}}{2} + C $$, where $$ C $$ is an arbitrary constant.
The integral of $$ (1-e^{x})^{2} $$ with respect to $$ x $$ is $$ x - 2e^{x} + \frac{e^{2x}}{2} + C $$.

6. \( \int\left(1-e^{x}\right)^{2} d x \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Calculus Questions