Find the integral $$ \int x^{5} \cos \left(x^{3}\right) d x= $$

Question

UpStudy AI · Accepted Answer

Calculate the integral $$ \int x^{5} \cos \left(x^{3}ight) d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int x^{5}\cos\left(x^{3}ight) dx$$
- step1: Use the substitution $$dx=\frac{1}{3x^{2}} dt$$ to transform the integral$$:$$
  $$\int x^{5}\cos\left(x^{3}ight)	imes \frac{1}{3x^{2}} dt$$
- step2: Simplify:
  $$\int \frac{x^{3}\cos\left(x^{3}ight)}{3} dt$$
- step3: Use the substitution $$t=x^{3}$$ to transform the integral$$:$$
  $$\int \frac{t\cos\left(tight)}{3} dt$$
- step4: Simplify the expression:
  $$\int \frac{1}{3}t\cos\left(tight) dt$$
- step5: Use properties of integrals:
  $$\frac{1}{3}	imes \int t\cos\left(tight) dt$$
- step6: Prepare for integration by parts:
  $$\begin{align}&u=t\&dv=\cos\left(tight)dt\end{align}$$
- step7: Calculate the derivative:
  $$\begin{align}&du=dt\&dv=\cos\left(tight)dt\end{align}$$
- step8: Evaluate the integral:
  $$\begin{align}&du=dt\&v=\sin\left(tight)\end{align}$$
- step9: Substitute the values into formula:
  $$\frac{1}{3}\left(t\sin\left(tight)-\int 1	imes \sin\left(tight) dtight)$$
- step10: Calculate:
  $$\frac{1}{3}\left(t\sin\left(tight)-\int \sin\left(tight) dtight)$$
- step11: Calculate:
  $$\frac{1}{3}t\sin\left(tight)-\frac{1}{3}	imes \int \sin\left(tight) dt$$
- step12: Evaluate the integral:
  $$\frac{1}{3}t\sin\left(tight)+\frac{1}{3}\cos\left(tight)$$
- step13: Substitute back:
  $$\frac{1}{3}x^{3}\sin\left(x^{3}ight)+\frac{1}{3}\cos\left(x^{3}ight)$$
- step14: Add the constant of integral C:
  $$\frac{1}{3}x^{3}\sin\left(x^{3}ight)+\frac{1}{3}\cos\left(x^{3}ight) + C, C \in \mathbb{R}$$
The integral of $$x^{5} \cos \left(x^{3}ight)$$ with respect to $$x$$ is $$\frac{1}{3}x^{3}\sin(x^{3}) + \frac{1}{3}\cos(x^{3}) + C$$, where $$C$$ is an arbitrary constant.
The integral of $$ x^{5} \cos \left(x^{3}ight) $$ with respect to $$ x $$ is $$ \frac{1}{3}x^{3}\sin(x^{3}) + \frac{1}{3}\cos(x^{3}) + C $$, where $$ C $$ is a constant.

Find the integral

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions