Calculus Examples

$\sin^{2} (2 t)$

Let $u_{1} = 2 t$ . Then $d u_{1} = 2 d t$ , so $\frac{1}{2} d u_{1} = d t$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Let $u_{1} = 2 t$ . Find $\frac{d u_{1}}{d t}$ .

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Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

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

$2 \cdot 1$

Multiply $2$ by $1$ .

$2$

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \sin^{2} (u_{1}) \frac{1}{2} d u_{1}$

Combine $\sin^{2} (u_{1})$ and $\frac{1}{2}$ .

$\int \frac{\sin^{2} (u_{1})}{2} d u_{1}$

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \sin^{2} (u_{1}) d u_{1}$

Use the half-angle formula to rewrite $\sin^{2} (u_{1})$ as $\frac{1 - \cos (2 u_{1})}{2}$ .

$\frac{1}{2} \int \frac{1 - \cos (2 u_{1})}{2} d u_{1}$

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} (\frac{1}{2} \int 1 - \cos (2 u_{1}) d u_{1})$

Simplify.

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Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{1}{2 \cdot 2} \int 1 - \cos (2 u_{1}) d u_{1}$

Multiply $2$ by $2$ .

$\frac{1}{4} \int 1 - \cos (2 u_{1}) d u_{1}$

Split the single integral into multiple integrals.

$\frac{1}{4} (\int d u_{1} + \int - \cos (2 u_{1}) d u_{1})$

Apply the constant rule.

$\frac{1}{4} (u_{1} + C + \int - \cos (2 u_{1}) d u_{1})$

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

$\frac{1}{4} (u_{1} + C - \int \cos (2 u_{1}) d u_{1})$

Let $u_{2} = 2 u_{1}$ . Then $d u_{2} = 2 d u_{1}$ , so $\frac{1}{2} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Let $u_{2} = 2 u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

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

$2 \cdot 1$

Multiply $2$ by $1$ .

$2$

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{4} (u_{1} + C - \int \cos (u_{2}) \frac{1}{2} d u_{2})$

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

$\frac{1}{4} (u_{1} + C - \int \frac{\cos (u_{2})}{2} d u_{2})$

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{4} (u_{1} + C - (\frac{1}{2} \int \cos (u_{2}) d u_{2}))$

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{4} (u_{1} + C - \frac{1}{2} (\sin (u_{2}) + C))$

Simplify.

$\frac{1}{4} (u_{1} - \frac{1}{2} \sin (u_{2})) + C$

Substitute back in for each integration substitution variable.

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Replace all occurrences of $u_{1}$ with $2 t$ .

$\frac{1}{4} (2 t - \frac{1}{2} \sin (u_{2})) + C$

Replace all occurrences of $u_{2}$ with $2 u_{1}$ .

$\frac{1}{4} (2 t - \frac{1}{2} \sin (2 u_{1})) + C$

Replace all occurrences of $u_{1}$ with $2 t$ .

$\frac{1}{4} (2 t - \frac{1}{2} \sin (2 (2 t))) + C$

Simplify.

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Simplify each term.

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Multiply $2$ by $2$ .

$\frac{1}{4} (2 t - \frac{1}{2} \sin (4 t)) + C$

Combine $\sin (4 t)$ and $\frac{1}{2}$ .

$\frac{1}{4} (2 t - \frac{\sin (4 t)}{2}) + C$

Apply the distributive property.

$\frac{1}{4} (2 t) + \frac{1}{4} (- \frac{\sin (4 t)}{2}) + C$

Cancel the common factor of $2$ .

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Factor $2$ out of $4$ .

$\frac{1}{2 (2)} (2 t) + \frac{1}{4} (- \frac{\sin (4 t)}{2}) + C$

Factor $2$ out of $2 t$ .

$\frac{1}{2 (2)} (2 (t)) + \frac{1}{4} (- \frac{\sin (4 t)}{2}) + C$

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 t) + \frac{1}{4} (- \frac{\sin (4 t)}{2}) + C$

Rewrite the expression.

$\frac{1}{2} t + \frac{1}{4} (- \frac{\sin (4 t)}{2}) + C$

Combine $\frac{1}{2}$ and $t$ .

$\frac{t}{2} + \frac{1}{4} (- \frac{\sin (4 t)}{2}) + C$

Multiply $\frac{1}{4} (- \frac{\sin (4 t)}{2})$ .

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Multiply $\frac{1}{4}$ by $\frac{\sin (4 t)}{2}$ .

$\frac{t}{2} - \frac{\sin (4 t)}{4 \cdot 2} + C$

Multiply $4$ by $2$ .

$\frac{t}{2} - \frac{\sin (4 t)}{8} + C$

Reorder terms.

$\frac{1}{2} t - \frac{1}{8} \sin (4 t) + C$