Calculus Examples

$\int_{1}^{x^{2}} \sqrt{1 + t^{3}} d t$

Step 1

Rewrite $1$ as $1^{3}$ .

$\frac{d}{d x} [\int_{1}^{x^{2}} \sqrt{1^{3} + t^{3}} d t]$

Step 2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = t$ .

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

Step 3

Simplify.

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Step 3.1

One to any power is one.

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

Step 3.2

Rewrite $- 1 t$ as $- t$ .

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

Step 4

Take the derivative of $\int_{1}^{x^{2}} \sqrt{(1 + t) (1 - t + t^{2})} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

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

Step 5

Differentiate using the Power Rule.

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Step 5.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x \sqrt{(1 + x^{2}) (1 - x^{2} + {(x^{2})}^{2})}$

Step 5.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 x \sqrt{(1 + x^{2}) (1 - x^{2} + x^{2 \cdot 2})}$

Step 5.2.2

Multiply $2$ by $2$ .

$2 x \sqrt{(1 + x^{2}) (1 - x^{2} + x^{4})}$