Calculus Examples

$\int \sqrt{x^{4} - 2 x^{3} + x^{2}} d x$

Step 1

Simplify.

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Step 1.1

Factor $x^{2}$ out of $x^{4} - 2 x^{3} + x^{2}$ .

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Step 1.1.1

Factor $x^{2}$ out of $x^{4}$ .

$\int \sqrt{x^{2} x^{2} - 2 x^{3} + x^{2}} d x$

Step 1.1.2

Factor $x^{2}$ out of $- 2 x^{3}$ .

$\int \sqrt{x^{2} x^{2} + x^{2} (- 2 x) + x^{2}} d x$

Step 1.1.3

Multiply by $1$ .

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

Step 1.1.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (- 2 x)$ .

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

Step 1.1.5

Factor $x^{2}$ out of $x^{2} (x^{2} - 2 x) + x^{2} \cdot 1$ .

$\int \sqrt{x^{2} (x^{2} - 2 x + 1)} d x$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $1$ as $1^{2}$ .

$\int \sqrt{x^{2} (x^{2} - 2 x + 1^{2})} d x$

Step 1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 1.2.3

Rewrite the polynomial.

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

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 1.3

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

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

Step 1.4

Pull terms out from under the radical, assuming positive real numbers.

$\int x (x - 1) d x$

Step 1.5

Apply the distributive property.

$\int x \cdot x + x \cdot - 1 d x$

Step 1.6

Multiply $x$ by $x$ .

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

Step 1.7

Move $- 1$ to the left of $x$ .

$\int x^{2} - 1 \cdot x d x$

Step 1.8

Rewrite $- 1 x$ as $- x$ .

$\int x^{2} - x d x$

Step 2

Split the single integral into multiple integrals.

$\int x^{2} d x + \int - x d x$

Step 3

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

$\frac{1}{3} x^{3} + C + \int - x d x$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{3} x^{3} + C - \int x d x$

Step 5

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

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

Step 6

Simplify.

$\frac{1}{3} x^{3} - \frac{1}{2} x^{2} + C$