Calculus Examples

$\int_{0}^{\sqrt[4]{π}} x^{3} \cos (x^{4}) d x$

Step 1

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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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2}$ .

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

Step 1.1.2

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

$2 x$

Step 1.2

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

$u_{lower} = 0^{2}$

Step 1.3

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0$

Step 1.4

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

$u_{upper} = {\sqrt[4]{π}}^{2}$

Step 1.5

Rewrite ${\sqrt[4]{π}}^{2}$ as $\sqrt{π}$ .

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Step 1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{π}$ as $π^{\frac{1}{4}}$ .

$u_{upper} = {(π^{\frac{1}{4}})}^{2}$

Step 1.5.2

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

$u_{upper} = π^{\frac{1}{4} \cdot 2}$

Step 1.5.3

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

$u_{upper} = π^{\frac{2}{4}}$

Step 1.5.4

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

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Step 1.5.4.1

Factor $2$ out of $2$ .

$u_{upper} = π^{\frac{2 (1)}{4}}$

Step 1.5.4.2

Cancel the common factors.

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Step 1.5.4.2.1

Factor $2$ out of $4$ .

$u_{upper} = π^{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 1.5.4.2.2

Cancel the common factor.

$u_{upper} = π^{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 1.5.4.2.3

Rewrite the expression.

$u_{upper} = π^{\frac{1}{2}}$

Step 1.5.5

Rewrite $π^{\frac{1}{2}}$ as $\sqrt{π}$ .

$u_{upper} = \sqrt{π}$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = \sqrt{π}$

Step 1.7

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

$\int_{0}^{\sqrt{π}} u_{1} \cos ({\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 2

Simplify.

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Step 2.1

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

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Step 2.1.1

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

$\int_{0}^{\sqrt{π}} u_{1} \cos ({({u_{1}}^{\frac{1}{2}})}^{4}) \frac{1}{2} d u_{1}$

Step 2.1.2

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

$\int_{0}^{\sqrt{π}} u_{1} \cos ({u_{1}}^{\frac{1}{2} \cdot 4}) \frac{1}{2} d u_{1}$

Step 2.1.3

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

$\int_{0}^{\sqrt{π}} u_{1} \cos ({u_{1}}^{\frac{4}{2}}) \frac{1}{2} d u_{1}$

Step 2.1.4

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

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Step 2.1.4.1

Factor $2$ out of $4$ .

$\int_{0}^{\sqrt{π}} u_{1} \cos ({u_{1}}^{\frac{2 \cdot 2}{2}}) \frac{1}{2} d u_{1}$

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $2$ out of $2$ .

$\int_{0}^{\sqrt{π}} u_{1} \cos ({u_{1}}^{\frac{2 \cdot 2}{2 (1)}}) \frac{1}{2} d u_{1}$

Step 2.1.4.2.2

Cancel the common factor.

$\int_{0}^{\sqrt{π}} u_{1} \cos ({u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}}) \frac{1}{2} d u_{1}$

Step 2.1.4.2.3

Rewrite the expression.

$\int_{0}^{\sqrt{π}} u_{1} \cos ({u_{1}}^{\frac{2}{1}}) \frac{1}{2} d u_{1}$

Step 2.1.4.2.4

Divide $2$ by $1$ .

$\int_{0}^{\sqrt{π}} u_{1} \cos ({u_{1}}^{2}) \frac{1}{2} d u_{1}$

Step 2.2

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

$\int_{0}^{\sqrt{π}} \frac{u_{1}}{2} \cos ({u_{1}}^{2}) d u_{1}$

Step 2.3

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

$\int_{0}^{\sqrt{π}} \frac{u_{1} \cos ({u_{1}}^{2})}{2} d u_{1}$

Step 3

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

$\frac{1}{2} \int_{0}^{\sqrt{π}} u_{1} \cos ({u_{1}}^{2}) d u_{1}$

Step 4

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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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

$2 u_{1}$

Step 4.2

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

$u_{lower} = 0^{2}$

Step 4.3

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0$

Step 4.4

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

$u_{upper} = {\sqrt{π}}^{2}$

Step 4.5

Rewrite ${\sqrt{π}}^{2}$ as $π$ .

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Step 4.5.1

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

$u_{upper} = {(π^{\frac{1}{2}})}^{2}$

Step 4.5.2

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

$u_{upper} = π^{\frac{1}{2} \cdot 2}$

Step 4.5.3

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

$u_{upper} = π^{\frac{2}{2}}$

Step 4.5.4

Cancel the common factor of $2$ .

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Step 4.5.4.1

Cancel the common factor.

$u_{upper} = π^{\frac{2}{2}}$

Step 4.5.4.2

Rewrite the expression.

$u_{upper} = π^{1}$

Step 4.5.5

Simplify.

$u_{upper} = π$

Step 4.6

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

$u_{lower} = 0$

$u_{upper} = π$

Step 4.7

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

$\frac{1}{2} \int_{0}^{π} \cos (u_{2}) \frac{1}{2} d u_{2}$

Step 5

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

$\frac{1}{2} \int_{0}^{π} \frac{\cos (u_{2})}{2} d u_{2}$

Step 6

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

$\frac{1}{2} (\frac{1}{2} \int_{0}^{π} \cos (u_{2}) d u_{2})$

Step 7

Simplify.

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Step 7.1

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

$\frac{1}{2 \cdot 2} \int_{0}^{π} \cos (u_{2}) d u_{2}$

Step 7.2

Multiply $2$ by $2$ .

$\frac{1}{4} \int_{0}^{π} \cos (u_{2}) d u_{2}$

Step 8

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

$\frac{1}{4} \sin (u_{2})]_{0}^{π}$

Step 9

Evaluate $\sin (u_{2})$ at $π$ and at $0$ .

$\frac{1}{4} (\sin (π) - \sin (0))$

Step 10

Simplify.

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Step 10.1

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{4} (\sin (π) - 0)$

Step 10.2

Multiply $- 1$ by $0$ .

$\frac{1}{4} (\sin (π) + 0)$

Step 10.3

Add $\sin (π)$ and $0$ .

$\frac{1}{4} \sin (π)$

Step 10.4

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

$\frac{\sin (π)}{4}$

Step 11

Simplify.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sin (0)}{4}$

Step 11.1.2

The exact value of $\sin (0)$ is $0$ .

$\frac{0}{4}$

Step 11.2

Divide $0$ by $4$ .

$0$