Calculus Examples

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

Step 1

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

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

Step 2

Differentiate.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.2

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

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

Step 2.3

Simplify terms.

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Step 2.3.1

Multiply $2$ by $1$ .

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

Step 2.3.2

Factor $2$ out of $2 x$ .

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

Step 2.3.3

Simplify the expression.

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Step 2.3.3.1

Apply the product rule to $2 (x)$ .

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

Step 2.3.3.2

Raise $2$ to the power of $2$ .

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

Step 2.3.4

Factor $2$ out of $2 x$ .

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

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Apply the product rule to $2 (x)$ .

$2 \frac{4 x^{2}}{2^{2} x^{2} + 1}$

Step 2.3.5.2

Raise $2$ to the power of $2$ .

$2 \frac{4 x^{2}}{4 x^{2} + 1}$

Step 2.3.6

Combine $2$ and $\frac{4 x^{2}}{4 x^{2} + 1}$ .

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

Step 2.3.7

Multiply $4$ by $2$ .

$\frac{8 x^{2}}{4 x^{2} + 1}$