Calculus Examples

$\frac{d u}{d x} = 7 x^{6} - 3 x^{2} + 8$

Step 1

Rewrite the equation.

$d u = (7 x^{6} - 3 x^{2} + 8) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d u = \int 7 x^{6} - 3 x^{2} + 8 d x$

Step 2.2

Apply the constant rule.

$u + C_{1} = \int 7 x^{6} - 3 x^{2} + 8 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$u + C_{1} = \int 7 x^{6} d x + \int - 3 x^{2} d x + \int 8 d x$

Step 2.3.2

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$u + C_{1} = 7 \int x^{6} d x + \int - 3 x^{2} d x + \int 8 d x$

Step 2.3.3

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$u + C_{1} = 7 (\frac{1}{7} x^{7} + C_{2}) + \int - 3 x^{2} d x + \int 8 d x$

Step 2.3.4

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

$u + C_{1} = 7 (\frac{1}{7} x^{7} + C_{2}) - 3 \int x^{2} d x + \int 8 d x$

Step 2.3.5

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

$u + C_{1} = 7 (\frac{1}{7} x^{7} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) + \int 8 d x$

Step 2.3.6

Apply the constant rule.

$u + C_{1} = 7 (\frac{1}{7} x^{7} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) + 8 x + C_{4}$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

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Step 2.3.7.1.1

Combine $\frac{1}{7}$ and $x^{7}$ .

$u + C_{1} = 7 (\frac{x^{7}}{7} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) + 8 x + C_{4}$

Step 2.3.7.1.2

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

$u + C_{1} = 7 (\frac{x^{7}}{7} + C_{2}) - 3 (\frac{x^{3}}{3} + C_{3}) + 8 x + C_{4}$

Step 2.3.7.2

Simplify.

$u + C_{1} = x^{7} - x^{3} + 8 x + C_{5}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$u = x^{7} - x^{3} + 8 x + K$