Integral of t²costdt dx

The solution

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$$\int\limits_{0}^{1} t^{2} \cos{\left(t \right)}\, dt$$

Integral(t^2*cos(t), (t, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(t \right)} = t^{2}$ and let $\operatorname{dv}{\left(t \right)} = \cos{\left(t \right)}$ .

Then $\operatorname{du}{\left(t \right)} = 2 t$ .

To find $v{\left(t \right)}$ :
1. The integral of cosine is sine:
  
  $\int \cos{\left(t \right)}\, dt = \sin{\left(t \right)}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(t \right)} = 2 t$ and let $\operatorname{dv}{\left(t \right)} = \sin{\left(t \right)}$ .

Then $\operatorname{du}{\left(t \right)} = 2$ .

To find $v{\left(t \right)}$ :
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(t \right)}\, dt = - \cos{\left(t \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- 2 \cos{\left(t \right)}\right)\, dt = - 2 \int \cos{\left(t \right)}\, dt$
1. The integral of cosine is sine:
  
  $\int \cos{\left(t \right)}\, dt = \sin{\left(t \right)}$
So, the result is: $- 2 \sin{\left(t \right)}$
Add the constant of integration:

$t^{2} \sin{\left(t \right)} + 2 t \cos{\left(t \right)} - 2 \sin{\left(t \right)}+ \mathrm{constant}$

The answer is:

$t^{2} \sin{\left(t \right)} + 2 t \cos{\left(t \right)} - 2 \sin{\left(t \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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 | t *cos(t) dt = C - 2*sin(t) + t *sin(t) + 2*t*cos(t)
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$$\int t^{2} \cos{\left(t \right)}\, dt = C + t^{2} \sin{\left(t \right)} + 2 t \cos{\left(t \right)} - 2 \sin{\left(t \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-sin(1) + 2*cos(1)

$$- \sin{\left(1 \right)} + 2 \cos{\left(1 \right)}$$

-sin(1) + 2*cos(1)

$$- \sin{\left(1 \right)} + 2 \cos{\left(1 \right)}$$

-sin(1) + 2*cos(1)

Numerical answer [src]

0.239133626928383

0.239133626928383

The graph

Use the examples entering the upper and lower limits of integration.

Other calculators

How to use it?

Identical expressions

Integral of t²costdt dx

Limits of integration:

The graph:

Piecewise:

The solution