Calculus Examples

\int \sqrt{1 - 4 x^{2}} d x

$\int \sqrt{1 - 4 x^{2}} d x$

Step 1

Let

x = \frac{1}{2} sin (t)

$x = \frac{1}{2} \sin (t)$ , where

- \frac{π}{2} \leq t \leq \frac{π}{2}

$- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then

d x = \frac{cos (t)}{2} d t

$d x = \frac{\cos (t)}{2} d t$ . Note that since

- \frac{π}{2} \leq t \leq \frac{π}{2}

$- \frac{π}{2} \leq t \leq \frac{π}{2}$ ,

\frac{cos (t)}{2}

$\frac{\cos (t)}{2}$ is positive.

\int \sqrt{1 - 4 {(\frac{1}{2} sin (t))}^{2}} \frac{cos (t)}{2} d t

$\int \sqrt{1 - 4 {(\frac{1}{2} \sin (t))}^{2}} \frac{\cos (t)}{2} d t$

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Simplify

\sqrt{1 - 4 {(\frac{1}{2} sin (t))}^{2}}

$\sqrt{1 - 4 {(\frac{1}{2} \sin (t))}^{2}}$ .

Tap for more steps...

Step 2.1.1

Simplify each term.

Tap for more steps...

Step 2.1.1.1

Combine

\frac{1}{2}

$\frac{1}{2}$ and

sin (t)

$\sin (t)$ .

\int \sqrt{1 - 4 {(\frac{sin (t)}{2})}^{2}} \frac{cos (t)}{2} d t

$\int \sqrt{1 - 4 {(\frac{\sin (t)}{2})}^{2}} \frac{\cos (t)}{2} d t$

Step 2.1.1.2

Apply the product rule to

\frac{sin (t)}{2}

$\frac{\sin (t)}{2}$ .

\int \sqrt{1 - 4 \frac{{sin}^{2} (t)}{2^{2}}} \frac{cos (t)}{2} d t

$\int \sqrt{1 - 4 \frac{\sin^{2} (t)}{2^{2}}} \frac{\cos (t)}{2} d t$

Step 2.1.1.3

Raise

2

$2$ to the power of

2

$2$ .

\int \sqrt{1 - 4 \frac{{sin}^{2} (t)}{4}} \frac{cos (t)}{2} d t

$\int \sqrt{1 - 4 \frac{\sin^{2} (t)}{4}} \frac{\cos (t)}{2} d t$

Step 2.1.1.4

Cancel the common factor of

4

$4$ .

Tap for more steps...

Step 2.1.1.4.1

Factor

4

$4$ out of

- 4

$- 4$ .

\int \sqrt{1 + 4 (- 1) \frac{{sin}^{2} (t)}{4}} \frac{cos (t)}{2} d t

$\int \sqrt{1 + 4 (- 1) \frac{\sin^{2} (t)}{4}} \frac{\cos (t)}{2} d t$

Step 2.1.1.4.2

Cancel the common factor.

$\int \sqrt{1 + 4 \cdot - 1 \frac{\sin^{2} (t)}{4}} \frac{\cos (t)}{2} d t$

Step 2.1.1.4.3

Rewrite the expression.

$\int \sqrt{1 - 1 \sin^{2} (t)} \frac{\cos (t)}{2} d t$

Step 2.1.1.5

Rewrite

$- 1 \sin^{2} (t)$ as

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

$\int \sqrt{1 - \sin^{2} (t)} \frac{\cos (t)}{2} d t$

Step 2.1.2

Apply pythagorean identity.

$\int \sqrt{\cos^{2} (t)} \frac{\cos (t)}{2} d t$

Step 2.1.3

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

$\int \cos (t) \frac{\cos (t)}{2} d t$

Step 2.2

Simplify.

Tap for more steps...

Step 2.2.1

Combine

$\cos (t)$ and

$\frac{\cos (t)}{2}$ .

$\int \frac{\cos (t) \cos (t)}{2} d t$

Step 2.2.2

Raise

$\cos (t)$ to the power of

$1$ .

$\int \frac{\cos^{1} (t) \cos (t)}{2} d t$

Step 2.2.3

Raise

$\cos (t)$ to the power of

$1$ .

$\int \frac{\cos^{1} (t) \cos^{1} (t)}{2} d t$

Step 2.2.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{{\cos (t)}^{1 + 1}}{2} d t$

Step 2.2.5

Add

$1$ and

$1$ .

$\int \frac{\cos^{2} (t)}{2} d t$

Step 3

Since

$\frac{1}{2}$ is constant with respect to

$t$ , move

$\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \cos^{2} (t) d t$

Step 4

Use the half-angle formula to rewrite

$\cos^{2} (t)$ as

$\frac{1 + \cos (2 t)}{2}$ .

$\frac{1}{2} \int \frac{1 + \cos (2 t)}{2} d t$

Step 5

Since

$\frac{1}{2}$ is constant with respect to

$t$ , move

$\frac{1}{2}$ out of the integral.

$\frac{1}{2} (\frac{1}{2} \int 1 + \cos (2 t) d t)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply

$\frac{1}{2}$ by

$\frac{1}{2}$ .

$\frac{1}{2 \cdot 2} \int 1 + \cos (2 t) d t$

Step 6.2

Multiply

$2$ by

$2$ .

$\frac{1}{4} \int 1 + \cos (2 t) d t$

Step 7

Split the single integral into multiple integrals.

$\frac{1}{4} (\int d t + \int \cos (2 t) d t)$

Step 8

Apply the constant rule.

$\frac{1}{4} (t + C + \int \cos (2 t) d t)$

Step 9

Let

$u = 2 t$ . Then

$d u = 2 d t$ , so

$\frac{1}{2} d u = d t$ . Rewrite using

$u$ and

$d$

$u$ .

Tap for more steps...

Step 9.1

Let

$u = 2 t$ . Find

$\frac{d u}{d t}$ .

Tap for more steps...

Step 9.1.1

Differentiate

$2 t$ .

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

Step 9.1.2

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 9.1.3

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 9.1.4

Multiply

$2$ by

$1$ .

$2$

Step 9.2

Rewrite the problem using

$u$ and

$d u$ .

$\frac{1}{4} (t + C + \int \cos (u) \frac{1}{2} d u)$

Step 10

Combine

$\cos (u)$ and

$\frac{1}{2}$ .

$\frac{1}{4} (t + C + \int \frac{\cos (u)}{2} d u)$

Step 11

Since

$\frac{1}{2}$ is constant with respect to

$u$ , move

$\frac{1}{2}$ out of the integral.

$\frac{1}{4} (t + C + \frac{1}{2} \int \cos (u) d u)$

Step 12

The integral of

$\cos (u)$ with respect to

$u$ is

$\sin (u)$ .

$\frac{1}{4} (t + C + \frac{1}{2} (\sin (u) + C))$

Step 13

Simplify.

$\frac{1}{4} (t + \frac{1}{2} \sin (u)) + C$

Step 14

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 14.1

Replace all occurrences of

$t$ with

$\arcsin (2 x)$ .

$\frac{1}{4} (\arcsin (2 x) + \frac{1}{2} \sin (u)) + C$

Step 14.2

Replace all occurrences of

$u$ with

$2 t$ .

$\frac{1}{4} (\arcsin (2 x) + \frac{1}{2} \sin (2 t)) + C$

Step 14.3

Replace all occurrences of

$t$ with

$\arcsin (2 x)$ .

$\frac{1}{4} (\arcsin (2 x) + \frac{1}{2} \sin (2 \arcsin (2 x))) + C$

Step 15

Simplify.

Tap for more steps...

Step 15.1

Combine

$\frac{1}{2}$ and

$\sin (2 \arcsin (2 x))$ .

$\frac{1}{4} (\arcsin (2 x) + \frac{\sin (2 \arcsin (2 x))}{2}) + C$

Step 15.2

Apply the distributive property.

$\frac{1}{4} \arcsin (2 x) + \frac{1}{4} \cdot \frac{\sin (2 \arcsin (2 x))}{2} + C$

Step 15.3

Combine

$\frac{1}{4}$ and

$\arcsin (2 x)$ .

$\frac{\arcsin (2 x)}{4} + \frac{1}{4} \cdot \frac{\sin (2 \arcsin (2 x))}{2} + C$

Step 15.4

Multiply

$\frac{1}{4} \cdot \frac{\sin (2 \arcsin (2 x))}{2}$ .

Tap for more steps...

Step 15.4.1

Multiply

$\frac{1}{4}$ by

$\frac{\sin (2 \arcsin (2 x))}{2}$ .

$\frac{\arcsin (2 x)}{4} + \frac{\sin (2 \arcsin (2 x))}{4 \cdot 2} + C$

Step 15.4.2

Multiply

$4$ by

$2$ .

$\frac{\arcsin (2 x)}{4} + \frac{\sin (2 \arcsin (2 x))}{8} + C$

Step 16

Reorder terms.

$\frac{1}{4} \arcsin (2 x) + \frac{1}{8} \sin (2 \arcsin (2 x)) + C$

Enter YOUR Problem