Calculus Examples

$\int_{0}^{x} \sqrt{t^{4} + t^{6}} d t$

Step 1

Factor $t^{4}$ out of $t^{4} + t^{6}$ .

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Step 1.1

Multiply by $1$ .

$\frac{d}{d x} [\int_{0}^{x} \sqrt{t^{4} \cdot 1 + t^{6}} d t]$

Step 1.2

Factor $t^{4}$ out of $t^{6}$ .

$\frac{d}{d x} [\int_{0}^{x} \sqrt{t^{4} \cdot 1 + t^{4} t^{2}} d t]$

Step 1.3

Factor $t^{4}$ out of $t^{4} \cdot 1 + t^{4} t^{2}$ .

$\frac{d}{d x} [\int_{0}^{x} \sqrt{t^{4} (1 + t^{2})} d t]$

Step 2

Rewrite $t^{4} (1 + t^{2})$ as ${(t^{2})}^{2} (1^{2} + t^{2})$ .

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Step 2.1

Rewrite $t^{4}$ as ${(t^{2})}^{2}$ .

$\frac{d}{d x} [\int_{0}^{x} \sqrt{{(t^{2})}^{2} (1 + t^{2})} d t]$

Step 2.2

Rewrite $1$ as $1^{2}$ .

$\frac{d}{d x} [\int_{0}^{x} \sqrt{{(t^{2})}^{2} (1^{2} + t^{2})} d t]$

Step 3

Simplify by cancelling exponent with radical.

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Step 3.1

Pull terms out from under the radical.

$\frac{d}{d x} [\int_{0}^{x} t^{2} \sqrt{1^{2} + t^{2}} d t]$

Step 3.2

One to any power is one.

$\frac{d}{d x} [\int_{0}^{x} t^{2} \sqrt{1 + t^{2}} d t]$

Step 4

Take the derivative of $\int_{0}^{x} t^{2} \sqrt{1 + t^{2}} d t$ with respect to $x$ using Fundamental Theorem of Calculus.

$x^{2} \sqrt{1 + x^{2}}$