Calculus Examples

$\int_{0}^{1} x \sin (π x^{2}) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $π x^{2}$ .

$\frac{d}{d x} [π x^{2}]$

Step 1.1.2

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

$π \frac{d}{d x} [x^{2}]$

Step 1.1.3

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

$π (2 x)$

Step 1.1.4

Move $2$ to the left of $π$ .

$2 π x$

Step 1.1.5

Reorder $2 π$ and $x$ .

$x (2 π)$

Step 1.1.6

Reorder $x$ and $2$ .

$2 \cdot x π$

Step 1.2

Substitute the lower limit in for $x$ in $u = π x^{2}$ .

$u_{lower} = π \cdot 0^{2}$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = π \cdot 0$

Step 1.3.2

Multiply $π$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = π x^{2}$ .

$u_{upper} = π \cdot 1^{2}$

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = π \cdot 1$

Step 1.5.2

Multiply $π$ by $1$ .

$u_{upper} = π$

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.

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

Step 2

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

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

Step 3

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

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

Step 4

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

$\frac{1}{2 π} - \cos (u)]_{0}^{π}$

Step 5

Evaluate $- \cos (u)$ at $π$ and at $0$ .

$\frac{1}{2 π} (- \cos (π) + \cos (0))$

Step 6

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{2 π} (- \cos (π) + 1)$

Step 7

Simplify.

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Step 7.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{1}{2 π} (- - \cos (0) + 1)$

Step 7.2

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{2 π} (- (- 1 \cdot 1) + 1)$

Step 7.3

Multiply $- 1$ by $1$ .

$\frac{1}{2 π} (- - 1 + 1)$

Step 7.4

Multiply $- 1$ by $- 1$ .

$\frac{1}{2 π} (1 + 1)$

Step 7.5

Add $1$ and $1$ .

$\frac{1}{2 π} \cdot 2$

Step 7.6

Cancel the common factor of $2$ .

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Step 7.6.1

Factor $2$ out of $2 π$ .

$\frac{1}{2 (π)} \cdot 2$

Step 7.6.2

Cancel the common factor.

$\frac{1}{2 π} \cdot 2$

Step 7.6.3

Rewrite the expression.

$\frac{1}{π}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{π}$

Decimal Form:

$0.31830988 \dots$