Calculus Examples

\int_{0}^{1} 6 x - 2 x^{3} - x \sqrt{x} + 3 e^{x} d x

$\int_{0}^{1} 6 x - 2 x^{3} - x \sqrt{x} + 3 e^{x} d x$

Step 1

Use

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

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

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

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

\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x \cdot x^{\frac{1}{2}} + 3 e^{x} d x]

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x \cdot x^{\frac{1}{2}} + 3 e^{x} d x]$

Step 2

Multiply

x

$x$ by

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ by adding the exponents.

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Step 2.1

Move

x^{\frac{1}{2}}

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

\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - (x^{\frac{1}{2}} x) + 3 e^{x} d x]

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - (x^{\frac{1}{2}} x) + 3 e^{x} d x]$

Step 2.2

Multiply

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ by

x

$x$ .

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Step 2.2.1

Raise

x

$x$ to the power of

1

$1$ .

\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - (x^{\frac{1}{2}} x^{1}) + 3 e^{x} d x]

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - (x^{\frac{1}{2}} x^{1}) + 3 e^{x} d x]$

Step 2.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

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

\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1}{2} + 1} + 3 e^{x} d x]

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1}{2} + 1} + 3 e^{x} d x]$

\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1}{2} + 1} + 3 e^{x} d x]

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1}{2} + 1} + 3 e^{x} d x]$

Step 2.3

Write

1

$1$ as a fraction with a common denominator.

\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1}{2} + \frac{2}{2}} + 3 e^{x} d x]

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1}{2} + \frac{2}{2}} + 3 e^{x} d x]$

Step 2.4

Combine the numerators over the common denominator.

\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1 + 2}{2}} + 3 e^{x} d x]

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1 + 2}{2}} + 3 e^{x} d x]$

Step 2.5

Add

1

$1$ and

2

$2$ .

\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x]

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x]$

\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x]

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x]$

Step 3

Once

\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x

$\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x$ has been evaluated, it will be constant with respect to

x

$x$ , so the derivative of

\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x

$\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x$ with respect to

x

$x$ is

0

$0$ .

0

$0$