Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t^{2}$ and $d v = \sin (B t)$ .

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

Step 2

Simplify.

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Step 2.1

Combine $t^{2}$ and $\frac{\cos (B t)}{B}$ .

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

Step 2.2

Multiply $2$ by $- 1$ .

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

Step 2.3

Combine $- 2$ and $\frac{\cos (B t)}{B}$ .

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

Step 2.4

Combine $\frac{- 2 \cos (B t)}{B}$ and $t$ .

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

Step 2.5

Move the negative in front of the fraction.

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

Step 3

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

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

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Multiply $\int \frac{2 \cos (B t) t}{B} d t$ by $1$ .

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

Step 5

Since $\frac{2}{B}$ is constant with respect to $t$ , move $\frac{2}{B}$ out of the integral.

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

Step 6

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t$ and $d v = \cos (B t)$ .

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

Step 7

Combine $t$ and $\frac{\sin (B t)}{B}$ .

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

Step 8

Since $\frac{1}{B}$ is constant with respect to $t$ , move $\frac{1}{B}$ out of the integral.

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

Step 9

Let $u = B t$ . Then $d u = B d t$ , so $\frac{1}{B} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 9.1

Let $u = B t$ . Find $\frac{d u}{d t}$ .

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Step 9.1.1

Differentiate $B t$ .

$\frac{d}{d t} [B t]$

Step 9.1.2

Since $B$ is constant with respect to $t$ , the derivative of $B t$ with respect to $t$ is $B \frac{d}{d t} [t]$ .

$B \frac{d}{d t} [t]$

Step 9.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$B \cdot 1$

Step 9.1.4

Multiply $B$ by $1$ .

$B$

Step 9.2

Rewrite the problem using $u$ and $d u$ .

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

Step 10

Combine $\sin (u)$ and $\frac{1}{B}$ .

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

Step 11

Since $\frac{1}{B}$ is constant with respect to $u$ , move $\frac{1}{B}$ out of the integral.

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

Step 12

Simplify.

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Step 12.1

Multiply $\frac{1}{B}$ by $\frac{1}{B}$ .

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

Step 12.2

Raise $B$ to the power of $1$ .

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

Step 12.3

Raise $B$ to the power of $1$ .

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

Step 12.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 12.5

Add $1$ and $1$ .

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

Step 13

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$- \frac{t^{2} \cos (B t)}{B} + \frac{2}{B} (\frac{t \sin (B t)}{B} - \frac{1}{B^{2}} (- \cos (u) + C))$

Step 14

Simplify.

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Step 14.1

Rewrite $- \frac{t^{2} \cos (B t)}{B} + \frac{2}{B} (\frac{t \sin (B t)}{B} - \frac{1}{B^{2}} (- \cos (u) + C))$ as $- \frac{t^{2} \cos (B t)}{B} + \frac{2}{B} (\frac{t \sin (B t)}{B} + \frac{\cos (u)}{B^{2}}) + C$ .

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

Step 14.2

Simplify.

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Step 14.2.1

To write $\frac{2}{B} (\frac{t \sin (B t)}{B} + \frac{\cos (u)}{B^{2}})$ as a fraction with a common denominator, multiply by $\frac{B}{B}$ .

$- \frac{t^{2} \cos (B t)}{B} + \frac{2}{B} (\frac{t \sin (B t)}{B} + \frac{\cos (u)}{B^{2}}) \cdot \frac{B}{B} + C$

Step 14.2.2

Combine $\frac{2}{B} (\frac{t \sin (B t)}{B} + \frac{\cos (u)}{B^{2}})$ and $\frac{B}{B}$ .

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

Step 14.2.3

Combine the numerators over the common denominator.

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

Step 14.2.4

Combine $B$ and $\frac{2}{B}$ .

$\frac{- t^{2} \cos (B t) + \frac{B \cdot 2}{B} (\frac{t \sin (B t)}{B} + \frac{\cos (u)}{B^{2}})}{B} + C$

Step 14.2.5

Move $2$ to the left of $B$ .

$\frac{- t^{2} \cos (B t) + \frac{2 \cdot B}{B} (\frac{t \sin (B t)}{B} + \frac{\cos (u)}{B^{2}})}{B} + C$

Step 14.2.6

Cancel the common factor of $B$ .

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Step 14.2.6.1

Cancel the common factor.

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

Step 14.2.6.2

Divide $2$ by $1$ .

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

Step 15

Replace all occurrences of $u$ with $B t$ .

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

Step 16

Simplify.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Apply the distributive property.

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

Step 16.1.2

Combine $2$ and $\frac{t \sin (B t)}{B}$ .

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

Step 16.1.3

Combine $2$ and $\frac{\cos (B t)}{B^{2}}$ .

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

Step 16.2

Factor $- 1$ out of $- t^{2} \cos (B t)$ .

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

Step 16.3

Factor $- 1$ out of $\frac{2 t \sin (B t)}{B}$ .

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

Step 16.4

Factor $- 1$ out of $- (t^{2} \cos (B t)) - \frac{- 2 t \sin (B t)}{B}$ .

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

Step 16.5

Factor $- 1$ out of $\frac{2 \cos (B t)}{B^{2}}$ .

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

Step 16.6

Factor $- 1$ out of $- (t^{2} \cos (B t) + \frac{- 2 t \sin (B t)}{B}) - \frac{- 2 \cos (B t)}{B^{2}}$ .

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

Step 16.7

Rewrite $- (t^{2} \cos (B t) + \frac{- 2 t \sin (B t)}{B} + \frac{- 2 \cos (B t)}{B^{2}})$ as $- 1 (t^{2} \cos (B t) + \frac{- 2 t \sin (B t)}{B} + \frac{- 2 \cos (B t)}{B^{2}})$ .

$\frac{- 1 (t^{2} \cos (B t) + \frac{- 2 t \sin (B t)}{B} + \frac{- 2 \cos (B t)}{B^{2}})}{B} + C$

Step 16.8

Move the negative in front of the fraction.

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

Step 16.9

Move the negative in front of the fraction.

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

Step 16.10

Move the negative in front of the fraction.

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