Calculus Examples

$\int \frac{- 20}{\sqrt{25 - 7 x^{2}}} d x$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Move the negative in front of the fraction.

$\int - \frac{20}{\sqrt{25 - 7 x^{2}}} d x$

Step 3

Since

$- 1$ is constant with respect to

$x$ , move

$- 1$ out of the integral.

$- \int \frac{20}{\sqrt{25 - 7 x^{2}}} d x$

Step 4

Since

$20$ is constant with respect to

$x$ , move

$20$ out of the integral.

$- (20 \int \frac{1}{\sqrt{25 - 7 x^{2}}} d x)$

Step 5

Multiply

$20$ by

$- 1$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 x^{2}}} d x$

Step 6

Let

$x = \frac{5}{\sqrt{7}} \sin (t)$ , where

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

$d x = \frac{5 \sqrt{7} \cos (t)}{7} d t$ . Note that since

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

$\frac{5 \sqrt{7} \cos (t)}{7}$ is positive.

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5}{\sqrt{7}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7

Simplify terms.

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Step 7.1

Simplify

$\sqrt{25 - 7 {(\frac{5}{\sqrt{7}} \sin (t))}^{2}}$ .

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

Multiply

$\frac{5}{\sqrt{7}}$ by

$\frac{\sqrt{7}}{\sqrt{7}}$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2

Combine and simplify the denominator.

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Step 7.1.1.2.1

Multiply

$\frac{5}{\sqrt{7}}$ by

$\frac{\sqrt{7}}{\sqrt{7}}$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{\sqrt{7} \sqrt{7}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2.2

Raise

$\sqrt{7}$ to the power of

$1$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2.3

Raise

$\sqrt{7}$ to the power of

$1$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2.4

Use the power rule

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

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{{\sqrt{7}}^{1 + 1}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2.5

Add

$1$ and

$1$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{{\sqrt{7}}^{2}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2.6

Rewrite

${\sqrt{7}}^{2}$ as

$7$ .

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Step 7.1.1.2.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{7}$ as

$7^{\frac{1}{2}}$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{7^{\frac{2}{2}}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2.6.4

Cancel the common factor of

$2$ .

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Step 7.1.1.2.6.4.1

Cancel the common factor.

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{7^{\frac{2}{2}}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2.6.4.2

Rewrite the expression.

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{7^{1}} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.2.6.5

Evaluate the exponent.

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7}}{7} \sin (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.3

Combine

$\frac{5 \sqrt{7}}{7}$ and

$\sin (t)$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 {(\frac{5 \sqrt{7} \sin (t)}{7})}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.4

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.1.1.4.1

Apply the product rule to

$\frac{5 \sqrt{7} \sin (t)}{7}$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{{(5 \sqrt{7} \sin (t))}^{2}}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.4.2

Apply the product rule to

$5 \sqrt{7} \sin (t)$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{{(5 \sqrt{7})}^{2} \sin^{2} (t)}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.4.3

Apply the product rule to

$5 \sqrt{7}$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{5^{2} {\sqrt{7}}^{2} \sin^{2} (t)}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.5

Simplify the numerator.

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Step 7.1.1.5.1

Raise

$5$ to the power of

$2$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{25 {\sqrt{7}}^{2} \sin^{2} (t)}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.5.2

Rewrite

${\sqrt{7}}^{2}$ as

$7$ .

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Step 7.1.1.5.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{7}$ as

$7^{\frac{1}{2}}$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{25 {(7^{\frac{1}{2}})}^{2} \sin^{2} (t)}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.5.2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{25 \cdot 7^{\frac{1}{2} \cdot 2} \sin^{2} (t)}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.5.2.3

Combine

$\frac{1}{2}$ and

$2$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{25 \cdot 7^{\frac{2}{2}} \sin^{2} (t)}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.5.2.4

Cancel the common factor of

$2$ .

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Step 7.1.1.5.2.4.1

Cancel the common factor.

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{25 \cdot 7^{\frac{2}{2}} \sin^{2} (t)}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.5.2.4.2

Rewrite the expression.

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{25 \cdot 7^{1} \sin^{2} (t)}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.5.2.5

Evaluate the exponent.

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{25 \cdot 7 \sin^{2} (t)}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.5.3

Multiply

$25$ by

$7$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{175 \sin^{2} (t)}{7^{2}}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.6

Raise

$7$ to the power of

$2$ .

$- 20 \int \frac{1}{\sqrt{25 - 7 \frac{175 \sin^{2} (t)}{49}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.7

Cancel the common factor of

$7$ .

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Step 7.1.1.7.1

Factor

$7$ out of

$- 7$ .

$- 20 \int \frac{1}{\sqrt{25 + 7 (- 1) \frac{175 \sin^{2} (t)}{49}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.7.2

Factor

$7$ out of

$49$ .

$- 20 \int \frac{1}{\sqrt{25 + 7 \cdot - 1 \frac{175 \sin^{2} (t)}{7 \cdot 7}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.7.3

Cancel the common factor.

$- 20 \int \frac{1}{\sqrt{25 + 7 \cdot - 1 \frac{175 \sin^{2} (t)}{7 \cdot 7}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.7.4

Rewrite the expression.

$- 20 \int \frac{1}{\sqrt{25 - 1 \frac{175 \sin^{2} (t)}{7}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.8

Cancel the common factor of

$175$ and

$7$ .

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Step 7.1.1.8.1

Factor

$7$ out of

$175 \sin^{2} (t)$ .

$- 20 \int \frac{1}{\sqrt{25 - 1 \frac{7 (25 \sin^{2} (t))}{7}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.8.2

Cancel the common factors.

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Step 7.1.1.8.2.1

Factor

$7$ out of

$7$ .

$- 20 \int \frac{1}{\sqrt{25 - 1 \frac{7 (25 \sin^{2} (t))}{7 (1)}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.8.2.2

Cancel the common factor.

$- 20 \int \frac{1}{\sqrt{25 - 1 \frac{7 (25 \sin^{2} (t))}{7 \cdot 1}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.8.2.3

Rewrite the expression.

$- 20 \int \frac{1}{\sqrt{25 - 1 \frac{25 \sin^{2} (t)}{1}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.8.2.4

Divide

$25 \sin^{2} (t)$ by

$1$ .

$- 20 \int \frac{1}{\sqrt{25 - 1 (25 \sin^{2} (t))}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.1.9

Multiply

$25$ by

$- 1$ .

$- 20 \int \frac{1}{\sqrt{25 - 25 \sin^{2} (t)}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.2

Factor

$25$ out of

$25$ .

$- 20 \int \frac{1}{\sqrt{25 (1) - 25 \sin^{2} (t)}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.3

Factor

$25$ out of

$- 25 \sin^{2} (t)$ .

$- 20 \int \frac{1}{\sqrt{25 (1) + 25 (- \sin^{2} (t))}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.4

Factor

$25$ out of

$25 (1) + 25 (- \sin^{2} (t))$ .

$- 20 \int \frac{1}{\sqrt{25 (1 - \sin^{2} (t))}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.5

Apply pythagorean identity.

$- 20 \int \frac{1}{\sqrt{25 \cos^{2} (t)}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.6

Rewrite

$25 \cos^{2} (t)$ as

${(5 \cos (t))}^{2}$ .

$- 20 \int \frac{1}{\sqrt{{(5 \cos (t))}^{2}}} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.1.7

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

$- 20 \int \frac{1}{5 \cos (t)} \cdot \frac{5 \sqrt{7} \cos (t)}{7} d t$

Step 7.2

Simplify.

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Step 7.2.1

Multiply

$\frac{1}{5 \cos (t)}$ by

$\frac{5 \sqrt{7} \cos (t)}{7}$ .

$- 20 \int \frac{5 \sqrt{7} \cos (t)}{5 \cos (t) \cdot 7} d t$

Step 7.2.2

Multiply

$7$ by

$5$ .

$- 20 \int \frac{5 \sqrt{7} \cos (t)}{35 \cos (t)} d t$

Step 7.2.3

Cancel the common factor of

$5$ and

$35$ .

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Step 7.2.3.1

Factor

$5$ out of

$5 \sqrt{7} \cos (t)$ .

$- 20 \int \frac{5 (\sqrt{7} \cos (t))}{35 \cos (t)} d t$

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor

$5$ out of

$35 \cos (t)$ .

$- 20 \int \frac{5 (\sqrt{7} \cos (t))}{5 (7 \cos (t))} d t$

Step 7.2.3.2.2

Cancel the common factor.

$- 20 \int \frac{5 (\sqrt{7} \cos (t))}{5 (7 \cos (t))} d t$

Step 7.2.3.2.3

Rewrite the expression.

$- 20 \int \frac{\sqrt{7} \cos (t)}{7 \cos (t)} d t$

Step 7.2.4

Cancel the common factor of

$\cos (t)$ .

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Step 7.2.4.1

Cancel the common factor.

$- 20 \int \frac{\sqrt{7} \cos (t)}{7 \cos (t)} d t$

Step 7.2.4.2

Rewrite the expression.

$- 20 \int \frac{\sqrt{7}}{7} d t$

Step 8

Apply the constant rule.

$- 20 (\frac{\sqrt{7}}{7} t + C)$

Step 9

Simplify the answer.

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Step 9.1

Rewrite

$- 20 (\frac{\sqrt{7}}{7} t + C)$ as

$- 20 \frac{\sqrt{7}}{7} t + C$ .

$- 20 \frac{\sqrt{7}}{7} t + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine

$- 20$ and

$\frac{\sqrt{7}}{7}$ .

$\frac{- 20 \sqrt{7}}{7} t + C$

Step 9.2.2

Move the negative in front of the fraction.

$- \frac{20 \sqrt{7}}{7} t + C$

Step 9.3

Replace all occurrences of

$t$ with

$\arcsin (\frac{\sqrt{7} x}{5})$ .

$- \frac{20 \sqrt{7}}{7} \arcsin (\frac{\sqrt{7} x}{5}) + C$

Step 9.4

Reorder terms.

$- \frac{20 \sqrt{7}}{7} \arcsin (\frac{\sqrt{7}}{5} x) + C$