$$ \int _{}^{}7(\cosh (x))^{3} d x $$

Question

UpStudy AI · Accepted Answer

It seems there was an issue with the simplification. Let's try to simplify the expression $$7(\cosh(x))^3$$ manually.

$$7(\cosh(x))^3 = 7(\frac{e^x + e^{-x}}{2})^3$$

Now, let's integrate the simplified expression.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int 7\left(\frac{e^{x}+e^{-x}}{2}ight)^{3} dx$$
- step1: Simplify the expression:
  $$\int 7\left(\frac{e^{x}}{2}+\frac{1}{2e^{x}}ight)^{3} dx$$
- step2: Use properties of integrals:
  $$7	imes \int \left(\frac{e^{x}}{2}+\frac{1}{2e^{x}}ight)^{3} dx$$
- step3: Evaluate the power:
  $$7	imes \int \left(\frac{e^{3x}}{8}+\frac{3e^{x}}{8}+\frac{3}{8e^{x}}+\frac{1}{8e^{3x}}ight) dx$$
- step4: Use properties of integrals:
  $$7\left(\int \frac{e^{3x}}{8} dx+\int \frac{3e^{x}}{8} dx+\int \frac{3}{8e^{x}} dx+\int \frac{1}{8e^{3x}} dxight)$$
- step5: Calculate:
  $$7	imes \int \frac{e^{3x}}{8} dx+7	imes \int \frac{3e^{x}}{8} dx+7	imes \int \frac{3}{8e^{x}} dx+7	imes \int \frac{1}{8e^{3x}} dx$$
- step6: Evaluate the integral:
  $$\frac{7e^{3x}}{24}+7	imes \int \frac{3e^{x}}{8} dx+7	imes \int \frac{3}{8e^{x}} dx+7	imes \int \frac{1}{8e^{3x}} dx$$
- step7: Evaluate the integral:
  $$\frac{7e^{3x}}{24}+\frac{21e^{x}}{8}+7	imes \int \frac{3}{8e^{x}} dx+7	imes \int \frac{1}{8e^{3x}} dx$$
- step8: Evaluate the integral:
  $$\frac{7e^{3x}}{24}+\frac{21e^{x}}{8}-\frac{21}{8e^{x}}+7	imes \int \frac{1}{8e^{3x}} dx$$
- step9: Evaluate the integral:
  $$\frac{7e^{3x}}{24}+\frac{21e^{x}}{8}-\frac{21}{8e^{x}}-\frac{7}{24e^{3x}}$$
- step10: Add the constant of integral C:
  $$\frac{7e^{3x}}{24}+\frac{21e^{x}}{8}-\frac{21}{8e^{x}}-\frac{7}{24e^{3x}} + C, C \in \mathbb{R}$$
After simplifying the expression $$7(\cosh(x))^3$$, the integral of the expression is:

$$\int 7(\cosh(x))^3 dx = \frac{7e^{3x}}{24} + \frac{21e^{x}}{8} - \frac{21}{8e^{x}} - \frac{7}{24e^{3x}} + C$$

where $$C$$ is the constant of integration.
$$ \int 7(\cosh(x))^3 dx = \frac{7e^{3x}}{24} + \frac{21e^{x}}{8} - \frac{21}{8e^{x}} - \frac{7}{24e^{3x}} + C $$

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions