Calculus Examples

$\int \frac{t - 2 t^{8}}{\sqrt{t}} d t$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\int \frac{t - 2 t^{8}}{t^{\frac{1}{2}}} d t$

Step 2

Simplify.

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Step 2.1

Factor $t$ out of $t - 2 t^{8}$ .

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Step 2.1.1

Raise $t$ to the power of $1$ .

$\int \frac{t^{1} - 2 t^{8}}{t^{\frac{1}{2}}} d t$

Step 2.1.2

Factor $t$ out of $t^{1}$ .

$\int \frac{t \cdot 1 - 2 t^{8}}{t^{\frac{1}{2}}} d t$

Step 2.1.3

Factor $t$ out of $- 2 t^{8}$ .

$\int \frac{t \cdot 1 + t (- 2 t^{7})}{t^{\frac{1}{2}}} d t$

Step 2.1.4

Factor $t$ out of $t \cdot 1 + t (- 2 t^{7})$ .

$\int \frac{t (1 - 2 t^{7})}{t^{\frac{1}{2}}} d t$

Step 2.2

Move $t^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\int t (1 - 2 t^{7}) t^{- \frac{1}{2}} d t$

Step 2.3

Multiply $t$ by $t^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.3.1

Move $t^{- \frac{1}{2}}$ .

$\int t^{- \frac{1}{2}} t (1 - 2 t^{7}) d t$

Step 2.3.2

Multiply $t^{- \frac{1}{2}}$ by $t$ .

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Step 2.3.2.1

Raise $t$ to the power of $1$ .

$\int t^{- \frac{1}{2}} t^{1} (1 - 2 t^{7}) d t$

Step 2.3.2.2

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

$\int t^{- \frac{1}{2} + 1} (1 - 2 t^{7}) d t$

Step 2.3.3

Write $1$ as a fraction with a common denominator.

$\int t^{- \frac{1}{2} + \frac{2}{2}} (1 - 2 t^{7}) d t$

Step 2.3.4

Combine the numerators over the common denominator.

$\int t^{\frac{- 1 + 2}{2}} (1 - 2 t^{7}) d t$

Step 2.3.5

Add $- 1$ and $2$ .

$\int t^{\frac{1}{2}} (1 - 2 t^{7}) d t$

Step 3

Expand $t^{\frac{1}{2}} (1 - 2 t^{7})$ .

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Step 3.1

Apply the distributive property.

$\int t^{\frac{1}{2}} \cdot 1 + t^{\frac{1}{2}} (- 2 t^{7}) d t$

Step 3.2

Reorder $t^{\frac{1}{2}}$ and $1$ .

$\int 1 \cdot t^{\frac{1}{2}} + t^{\frac{1}{2}} (- 2 t^{7}) d t$

Step 3.3

Reorder $t^{\frac{1}{2}}$ and $- 2$ .

$\int 1 \cdot t^{\frac{1}{2}} - 2 \cdot t^{\frac{1}{2}} t^{7} d t$

Step 3.4

Multiply $t^{\frac{1}{2}}$ by $1$ .

$\int t^{\frac{1}{2}} - 2 t^{\frac{1}{2}} t^{7} d t$

Step 3.5

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

$\int t^{\frac{1}{2}} - 2 t^{\frac{1}{2} + 7} d t$

Step 3.6

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int t^{\frac{1}{2}} - 2 t^{\frac{1}{2} + 7 \cdot \frac{2}{2}} d t$

Step 3.7

Combine $7$ and $\frac{2}{2}$ .

$\int t^{\frac{1}{2}} - 2 t^{\frac{1}{2} + \frac{7 \cdot 2}{2}} d t$

Step 3.8

Combine the numerators over the common denominator.

$\int t^{\frac{1}{2}} - 2 t^{\frac{1 + 7 \cdot 2}{2}} d t$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $7$ by $2$ .

$\int t^{\frac{1}{2}} - 2 t^{\frac{1 + 14}{2}} d t$

Step 3.9.2

Add $1$ and $14$ .

$\int t^{\frac{1}{2}} - 2 t^{\frac{15}{2}} d t$

Step 4

Split the single integral into multiple integrals.

$\int t^{\frac{1}{2}} d t + \int - 2 t^{\frac{15}{2}} d t$

Step 5

By the Power Rule, the integral of $t^{\frac{1}{2}}$ with respect to $t$ is $\frac{2}{3} t^{\frac{3}{2}}$ .

$\frac{2}{3} t^{\frac{3}{2}} + C + \int - 2 t^{\frac{15}{2}} d t$

Step 6

Since $- 2$ is constant with respect to $t$ , move $- 2$ out of the integral.

$\frac{2}{3} t^{\frac{3}{2}} + C - 2 \int t^{\frac{15}{2}} d t$

Step 7

By the Power Rule, the integral of $t^{\frac{15}{2}}$ with respect to $t$ is $\frac{2}{17} t^{\frac{17}{2}}$ .

$\frac{2}{3} t^{\frac{3}{2}} + C - 2 (\frac{2}{17} t^{\frac{17}{2}} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{2}{3} t^{\frac{3}{2}} - 2 (\frac{2}{17}) t^{\frac{17}{2}} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $- 2$ and $\frac{2}{17}$ .

$\frac{2}{3} t^{\frac{3}{2}} + \frac{- 2 \cdot 2}{17} t^{\frac{17}{2}} + C$

Step 8.2.2

Multiply $- 2$ by $2$ .

$\frac{2}{3} t^{\frac{3}{2}} + \frac{- 4}{17} t^{\frac{17}{2}} + C$

Step 8.2.3

Move the negative in front of the fraction.

$\frac{2}{3} t^{\frac{3}{2}} - \frac{4}{17} t^{\frac{17}{2}} + C$