Calculus Examples

$\int_{0}^{x} \sqrt{t + t^{3}} d t$

Step 1

Factor $t$ out of $t + t^{3}$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Factor $t$ out of $t^{3}$ .

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

Step 1.4

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

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

Step 2

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

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