Calculus Examples

$t (t^{2} + 1)$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int t \cdot t^{2} + t \cdot 1 d t$

Step 1.2

Reorder $t$ and $1$ .

$\int t \cdot t^{2} + 1 \cdot t d t$

Step 1.3

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

$\int t^{1} t^{2} + 1 t d t$

Step 1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int t^{1 + 2} + 1 t d t$

Step 1.5

Add $1$ and $2$ .

$\int t^{3} + 1 t d t$

Step 1.6

Multiply $t$ by $1$ .

$\int t^{3} + t d t$

Step 2

Split the single integral into multiple integrals.

$\int t^{3} d t + \int t d t$

Step 3

By the Power Rule, the integral of $t^{3}$ with respect to $t$ is $\frac{1}{4} t^{4}$ .

$\frac{1}{4} t^{4} + C + \int t d t$

Step 4

By the Power Rule, the integral of $t$ with respect to $t$ is $\frac{1}{2} t^{2}$ .

$\frac{1}{4} t^{4} + C + \frac{1}{2} t^{2} + C$

Step 5

Simplify.

$\frac{1}{4} t^{4} + \frac{1}{2} t^{2} + C$