Calculus Examples

$\int_{1}^{x^{3}} \frac{1}{1 + \ln (t)} d t$

Step 1

Take the derivative of $\int_{1}^{x^{3}} \frac{1}{1 + \ln (t)} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [x^{3}] \frac{1}{1 + \ln (x^{3})}$

Step 2

Differentiate using the Power Rule.

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Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} \frac{1}{1 + \ln (x^{3})}$

Step 2.2

Combine fractions.

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Step 2.2.1

Combine $3$ and $\frac{1}{1 + \ln (x^{3})}$ .

$x^{2} \frac{3}{1 + \ln (x^{3})}$

Step 2.2.2

Combine $x^{2}$ and $\frac{3}{1 + \ln (x^{3})}$ .

$\frac{x^{2} \cdot 3}{1 + \ln (x^{3})}$

Step 2.2.3

Move $3$ to the left of $x^{2}$ .

$\frac{3 x^{2}}{1 + \ln (x^{3})}$