Calculus Examples

\int t^{2} \cos (t) d t

$\int t^{2} \cos (t) d t$

Step 1

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = t^{2}

$u = t^{2}$ and

d v = \cos (t)

$d v = \cos (t)$ .

t^{2} \sin (t) - \int \sin (t) (2 t) d t

$t^{2} \sin (t) - \int \sin (t) (2 t) d t$

Step 2

Since

2

$2$ is constant with respect to

t

$t$ , move

2

$2$ out of the integral.

t^{2} \sin (t) - (2 \int \sin (t) (t) d t)

$t^{2} \sin (t) - (2 \int \sin (t) (t) d t)$

Step 3

Multiply

2

$2$ by

- 1

$- 1$ .

t^{2} \sin (t) - 2 \int \sin (t) (t) d t

$t^{2} \sin (t) - 2 \int \sin (t) (t) d t$

Step 4

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = t

$u = t$ and

d v = \sin (t)

$d v = \sin (t)$ .

t^{2} \sin (t) - 2 (t (- \cos (t)) - \int - \cos (t) d t)

$t^{2} \sin (t) - 2 (t (- \cos (t)) - \int - \cos (t) d t)$

Step 5

Since

- 1

$- 1$ is constant with respect to

t

$t$ , move

- 1

$- 1$ out of the integral.

t^{2} \sin (t) - 2 (t (- \cos (t)) - - \int \cos (t) d t)

$t^{2} \sin (t) - 2 (t (- \cos (t)) - - \int \cos (t) d t)$

Step 6

Simplify.

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Step 6.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

t^{2} \sin (t) - 2 (t (- \cos (t)) + 1 \int \cos (t) d t)

$t^{2} \sin (t) - 2 (t (- \cos (t)) + 1 \int \cos (t) d t)$

Step 6.2

Multiply

\int \cos (t) d t

$\int \cos (t) d t$ by

1

$1$ .

t^{2} \sin (t) - 2 (t (- \cos (t)) + \int \cos (t) d t)

$t^{2} \sin (t) - 2 (t (- \cos (t)) + \int \cos (t) d t)$

t^{2} \sin (t) - 2 (t (- \cos (t)) + \int \cos (t) d t)

$t^{2} \sin (t) - 2 (t (- \cos (t)) + \int \cos (t) d t)$

Step 7

The integral of

\cos (t)

$\cos (t)$ with respect to

t

$t$ is

\sin (t)

$\sin (t)$ .

t^{2} \sin (t) - 2 (t (- \cos (t)) + \sin (t) + C)

$t^{2} \sin (t) - 2 (t (- \cos (t)) + \sin (t) + C)$

Step 8

Rewrite

t^{2} \sin (t) - 2 (t (- \cos (t)) + \sin (t) + C)

$t^{2} \sin (t) - 2 (t (- \cos (t)) + \sin (t) + C)$ as

t^{2} \sin (t) - 2 (- t \cos (t) + \sin (t)) + C

$t^{2} \sin (t) - 2 (- t \cos (t) + \sin (t)) + C$ .

t^{2} \sin (t) - 2 (- t \cos (t) + \sin (t)) + C

$t^{2} \sin (t) - 2 (- t \cos (t) + \sin (t)) + C$