Calculus Examples

$\int_{0}^{x} \frac{t^{2}}{1 + t^{3}} d t$

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

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 $b = t$ .

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

Step 1.3

Simplify.

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Step 1.3.1

One to any power is one.

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

Step 1.3.2

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

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

Step 2

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

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