Calculus Examples

$\int_{- 1}^{1} (1 + \sqrt{1 - x^{2}}) d x$

Step 1

Remove parentheses.

$\int_{- 1}^{1} 1 + \sqrt{1 - x^{2}} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 1}^{1} d x + \int_{- 1}^{1} \sqrt{1 - x^{2}} d x$

Step 3

Apply the constant rule.

$x]_{- 1}^{1} + \int_{- 1}^{1} \sqrt{1 - x^{2}} d x$

Step 4

Let $x = \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\cos (t)$ is positive.

$x]_{- 1}^{1} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{1 - \sin^{2} (t)} \cos (t) d t$

Step 5

Simplify terms.

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Step 5.1

Simplify $\sqrt{1 - \sin^{2} (t)}$ .

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Step 5.1.1

Apply pythagorean identity.

$x]_{- 1}^{1} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{\cos^{2} (t)} \cos (t) d t$

Step 5.1.2

Pull terms out from under the radical, assuming positive real numbers.

$x]_{- 1}^{1} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (t) \cos (t) d t$

Step 5.2

Simplify.

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Step 5.2.1

Raise $\cos (t)$ to the power of $1$ .

$x]_{- 1}^{1} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{1} (t) \cos (t) d t$

Step 5.2.2

Raise $\cos (t)$ to the power of $1$ .

$x]_{- 1}^{1} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{1} (t) \cos^{1} (t) d t$

Step 5.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x]_{- 1}^{1} + \int_{- \frac{π}{2}}^{\frac{π}{2}} {\cos (t)}^{1 + 1} d t$

Step 5.2.4

Add $1$ and $1$ .

$x]_{- 1}^{1} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{2} (t) d t$

Step 6

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

$x]_{- 1}^{1} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + \cos (2 t)}{2} d t$

Step 7

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

$x]_{- 1}^{1} + \frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 8

Split the single integral into multiple integrals.

$x]_{- 1}^{1} + \frac{1}{2} (\int_{- \frac{π}{2}}^{\frac{π}{2}} d t + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (2 t) d t)$

Step 9

Apply the constant rule.

$x]_{- 1}^{1} + \frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (2 t) d t)$

Step 10

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

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Step 10.1

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

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Step 10.1.1

Differentiate $2 t$ .

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

Step 10.1.2

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

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

Step 10.1.3

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

$2 \cdot 1$

Step 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

Substitute the lower limit in for $t$ in $u = 2 t$ .

$u_{lower} = 2 (- \frac{π}{2})$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$u_{lower} = 2 \frac{- π}{2}$

Step 10.3.2

Cancel the common factor.

$u_{lower} = 2 \frac{- π}{2}$

Step 10.3.3

Rewrite the expression.

$u_{lower} = - π$

Step 10.4

Substitute the upper limit in for $t$ in $u = 2 t$ .

$u_{upper} = 2 \frac{π}{2}$

Step 10.5

Cancel the common factor of $2$ .

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Step 10.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 10.5.2

Rewrite the expression.

$u_{upper} = π$

Step 10.6

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

$u_{lower} = - π$

$u_{upper} = π$

Step 10.7

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

$x]_{- 1}^{1} + \frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- π}^{π} \cos (u) \frac{1}{2} d u)$

Step 11

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

$x]_{- 1}^{1} + \frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- π}^{π} \frac{\cos (u)}{2} d u)$

Step 12

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

$x]_{- 1}^{1} + \frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{1}{2} \int_{- π}^{π} \cos (u) d u)$

Step 13

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

$x]_{- 1}^{1} + \frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{1}{2} \sin (u)]_{- π}^{π})$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $x$ at $1$ and at $- 1$ .

$(1) + 1 + \frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{1}{2} \sin (u)]_{- π}^{π})$

Step 14.2

Evaluate $t$ at $\frac{π}{2}$ and at $- \frac{π}{2}$ .

$(1) + 1 + \frac{1}{2} ((\frac{π}{2}) + \frac{π}{2} + \frac{1}{2} \sin (u)]_{- π}^{π})$

Step 14.3

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

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

Step 14.4

Simplify.

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Step 14.4.1

Add $1$ and $1$ .

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

Step 14.4.2

Combine the numerators over the common denominator.

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

Step 14.4.3

Add $π$ and $π$ .

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

Step 14.4.4

Cancel the common factor of $2$ .

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Step 14.4.4.1

Cancel the common factor.

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

Step 14.4.4.2

Divide $π$ by $1$ .

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

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

Step 15

Simplify each term.

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Step 15.1

Simplify each term.

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Step 15.1.1

Simplify each term.

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Step 15.1.1.1

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

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

Step 15.1.1.2

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

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

Step 15.1.1.3

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

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

Step 15.1.1.4

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

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

Step 15.1.1.5

Multiply $- 1$ by $0$ .

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

Step 15.1.2

Add $0$ and $0$ .

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

Step 15.1.3

Multiply $\frac{1}{2}$ by $0$ .

$2 + \frac{1}{2} (π + 0)$

Step 15.2

Add $π$ and $0$ .

$2 + \frac{1}{2} π$

Step 15.3

Combine $\frac{1}{2}$ and $π$ .

$2 + \frac{π}{2}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$2 + \frac{π}{2}$

Decimal Form:

$3.57079632 \dots$

Step 17