Calculus Examples

$\int (1 + 3 t) t^{2} d t$

Step 1

Expand $(1 + 3 t) t^{2}$ .

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Step 1.1

Apply the distributive property.

$\int 1 t^{2} + 3 t \cdot t^{2} d t$

Step 1.2

Multiply $t^{2}$ by $1$ .

$\int t^{2} + 3 t \cdot t^{2} d t$

Step 1.3

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

$\int t^{2} + 3 (t^{1} t^{2}) d t$

Step 1.4

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

$\int t^{2} + 3 t^{1 + 2} d t$

Step 1.5

Add $1$ and $2$ .

$\int t^{2} + 3 t^{3} d t$

Step 1.6

Reorder $t^{2}$ and $3 t^{3}$ .

$\int 3 t^{3} + t^{2} d t$

Step 2

Split the single integral into multiple integrals.

$\int 3 t^{3} d t + \int t^{2} d t$

Step 3

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

$3 \int t^{3} d t + \int t^{2} d t$

Step 4

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

$3 (\frac{1}{4} t^{4} + C) + \int t^{2} d t$

Step 5

By the Power Rule, the integral of $t^{2}$ with respect to $t$ is $\frac{1}{3} t^{3}$ .

$3 (\frac{1}{4} t^{4} + C) + \frac{1}{3} t^{3} + C$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{4}$ and $t^{4}$ .

$3 (\frac{t^{4}}{4} + C) + \frac{1}{3} t^{3} + C$

Step 6.2

Simplify.

$\frac{3 t^{4}}{4} + \frac{1}{3} t^{3} + C$

Step 6.3

Reorder terms.

$\frac{3}{4} t^{4} + \frac{1}{3} t^{3} + C$