Calculus Examples

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

Step 1

Expand ${(2 t^{2} - 1)}^{2}$ .

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Step 1.1

Rewrite ${(2 t^{2} - 1)}^{2}$ as $(2 t^{2} - 1) (2 t^{2} - 1)$ .

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Move $t^{2}$ .

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

Step 1.6

Move $t^{2}$ .

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

Step 1.7

Multiply $2$ by $2$ .

$\int 4 t^{2} t^{2} + 2 \cdot - 1 t^{2} - 1 \cdot 2 t^{2} - 1 \cdot - 1 d t$

Step 1.8

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

$\int 4 t^{2 + 2} + 2 \cdot - 1 t^{2} - 1 \cdot 2 t^{2} - 1 \cdot - 1 d t$

Step 1.9

Add $2$ and $2$ .

$\int 4 t^{4} + 2 \cdot - 1 t^{2} - 1 \cdot 2 t^{2} - 1 \cdot - 1 d t$

Step 1.10

Multiply $2$ by $- 1$ .

$\int 4 t^{4} - 2 t^{2} - 1 \cdot 2 t^{2} - 1 \cdot - 1 d t$

Step 1.11

Multiply $- 1$ by $2$ .

$\int 4 t^{4} - 2 t^{2} - 2 t^{2} - 1 \cdot - 1 d t$

Step 1.12

Multiply $- 1$ by $- 1$ .

$\int 4 t^{4} - 2 t^{2} - 2 t^{2} + 1 d t$

Step 1.13

Subtract $2 t^{2}$ from $- 2 t^{2}$ .

$\int 4 t^{4} - 4 t^{2} + 1 d t$

Step 2

Split the single integral into multiple integrals.

$\int 4 t^{4} d t + \int - 4 t^{2} d t + \int d t$

Step 3

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$4 \int t^{4} d t + \int - 4 t^{2} d t + \int d t$

Step 4

By the Power Rule, the integral of $t^{4}$ with respect to $t$ is $\frac{1}{5} t^{5}$ .

$4 (\frac{1}{5} t^{5} + C) + \int - 4 t^{2} d t + \int d t$

Step 5

Since $- 4$ is constant with respect to $t$ , move $- 4$ out of the integral.

$4 (\frac{1}{5} t^{5} + C) - 4 \int t^{2} d t + \int d t$

Step 6

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

$4 (\frac{1}{5} t^{5} + C) - 4 (\frac{1}{3} t^{3} + C) + \int d t$

Step 7

Apply the constant rule.

$4 (\frac{1}{5} t^{5} + C) - 4 (\frac{1}{3} t^{3} + C) + t + C$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{5}$ and $t^{5}$ .

$4 (\frac{t^{5}}{5} + C) - 4 (\frac{1}{3} t^{3} + C) + t + C$

Step 8.1.2

Combine $\frac{1}{3}$ and $t^{3}$ .

$4 (\frac{t^{5}}{5} + C) - 4 (\frac{t^{3}}{3} + C) + t + C$

Step 8.2

Simplify.

$\frac{4 t^{5}}{5} - \frac{4 t^{3}}{3} + t + C$

Step 9

Reorder terms.

$\frac{4}{5} t^{5} - \frac{4}{3} t^{3} + t + C$