Calculus Examples

$\int_{0}^{\sqrt{π}} x \cos (x^{2}) d x$

Step 1

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

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Step 1.1

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

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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 $u = x^{2}$ .

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

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 $u = x^{2}$ .

$u_{upper} = {\sqrt{π}}^{2}$

Step 1.5

Rewrite ${\sqrt{π}}^{2}$ as $π$ .

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Step 1.5.1

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 1.5.2

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

$u_{upper} = π^{\frac{1}{2} \cdot 2}$

Step 1.5.3

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

$u_{upper} = π^{\frac{2}{2}}$

Step 1.5.4

Cancel the common factor of $2$ .

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Step 1.5.4.1

Cancel the common factor.

$u_{upper} = π^{\frac{2}{2}}$

Step 1.5.4.2

Rewrite the expression.

$u_{upper} = π^{1}$

Step 1.5.5

Simplify.

$u_{upper} = π$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = π$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

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

Step 2

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

$\int_{0}^{π} \frac{\cos (u)}{2} d u$

Step 3

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

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

Step 4

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

$\frac{1}{2} \sin (u)]_{0}^{π}$

Step 5

Evaluate $\sin (u)$ at $π$ and at $0$ .

$\frac{1}{2} (\sin (π) - \sin (0))$

Step 6

Simplify.

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Step 6.1

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

$\frac{1}{2} (\sin (π) - 0)$

Step 6.2

Multiply $- 1$ by $0$ .

$\frac{1}{2} (\sin (π) + 0)$

Step 6.3

Add $\sin (π)$ and $0$ .

$\frac{1}{2} \sin (π)$

Step 6.4

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

$\frac{\sin (π)}{2}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{\sin (0)}{2}$

Step 7.1.2

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

$\frac{0}{2}$

Step 7.2

Divide $0$ by $2$ .

$0$