Calculus Examples

$\int_{0.4}^{1} x^{3} \cot (x^{4}) d x$

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

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

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Differentiate $x^{2}$ .

$\frac{d}{d x} [x^{2}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x$

Substitute the lower limit in for $x$ in $u_{1} = x^{2}$ .

$u_{lower} = {0.4}^{2}$

Raise $0.4$ to the power of $2$ .

$u_{lower} = 0.16$

Substitute the upper limit in for $x$ in $u_{1} = x^{2}$ .

$u_{upper} = 1^{2}$

One to any power is one.

$u_{upper} = 1$

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0.16 u_{upper} = 1$

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$\int_{0.16}^{1} u_{1} \cot ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Simplify.

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Rewrite ${\sqrt{u_{1}}}^{4}$ as ${u_{1}}^{2}$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u_{1}}$ as ${u_{1}}^{\frac{1}{2}}$ .

$\int_{0.16}^{1} u_{1} \cot ({({u_{1}}^{\frac{1}{2}})}^{4}) \frac{1}{2} d u_{1}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int_{0.16}^{1} u_{1} \cot ({u_{1}}^{\frac{1}{2} \cdot 4}) \frac{1}{2} d u_{1}$

Combine $\frac{1}{2}$ and $4$ .

$\int_{0.16}^{1} u_{1} \cot ({u_{1}}^{\frac{4}{2}}) \frac{1}{2} d u_{1}$

Cancel the common factor of $4$ and $2$ .

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Factor $2$ out of $4$ .

$\int_{0.16}^{1} u_{1} \cot ({u_{1}}^{\frac{2 \cdot 2}{2}}) \frac{1}{2} d u_{1}$

Cancel the common factors.

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Factor $2$ out of $2$ .

$\int_{0.16}^{1} u_{1} \cot ({u_{1}}^{\frac{2 \cdot 2}{2 (1)}}) \frac{1}{2} d u_{1}$

Cancel the common factor.

$\int_{0.16}^{1} u_{1} \cot ({u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}}) \frac{1}{2} d u_{1}$

Rewrite the expression.

$\int_{0.16}^{1} u_{1} \cot ({u_{1}}^{\frac{2}{1}}) \frac{1}{2} d u_{1}$

Divide $2$ by $1$ .

$\int_{0.16}^{1} u_{1} \cot ({u_{1}}^{2}) \frac{1}{2} d u_{1}$

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

$\int_{0.16}^{1} \frac{u_{1}}{2} \cot ({u_{1}}^{2}) d u_{1}$

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

$\int_{0.16}^{1} \frac{u_{1} \cot ({u_{1}}^{2})}{2} d u_{1}$

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

$\frac{1}{2} \int_{0.16}^{1} u_{1} \cot ({u_{1}}^{2}) d u_{1}$

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

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

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Differentiate ${u_{1}}^{2}$ .

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

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1}$

Substitute the lower limit in for $u_{1}$ in $u_{2} = {u_{1}}^{2}$ .

$u_{lower} = {0.16}^{2}$

Raise $0.16$ to the power of $2$ .

$u_{lower} = 0.0256$

Substitute the upper limit in for $u_{1}$ in $u_{2} = {u_{1}}^{2}$ .

$u_{upper} = 1^{2}$

One to any power is one.

$u_{upper} = 1$

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0.0256 u_{upper} = 1$

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\frac{1}{2} \int_{0.0256}^{1} \cot (u_{2}) \frac{1}{2} d u_{2}$

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

$\frac{1}{2} \int_{0.0256}^{1} \frac{\cot (u_{2})}{2} d u_{2}$

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

$\frac{1}{2} (\frac{1}{2} \int_{0.0256}^{1} \cot (u_{2}) d u_{2})$

Simplify.

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

$\frac{1}{2 \cdot 2} \int_{0.0256}^{1} \cot (u_{2}) d u_{2}$

Multiply $2$ by $2$ .

$\frac{1}{4} \int_{0.0256}^{1} \cot (u_{2}) d u_{2}$

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

$\frac{1}{4} \ln (| \sin (u_{2}) |)]_{0.0256}^{1}$

Evaluate $\ln (| \sin (u_{2}) |)$ at $1$ and at $0.0256$ .

$\frac{1}{4} (\ln (| \sin (1) |) - \ln (| \sin (0.0256) |))$

Simplify.

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Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{1}{4} \ln (\frac{| \sin (1) |}{| \sin (0.0256) |})$

Combine $\frac{1}{4}$ and $\ln (\frac{| \sin (1) |}{| \sin (0.0256) |})$ .

$\frac{\ln (\frac{| \sin (1) |}{| \sin (0.0256) |})}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (\frac{| \sin (1) |}{| \sin (0.0256) |})}{4}$

Decimal Form:

$- 0.09575139 \dots$