Calculus Examples

$\frac{d u}{d x} = 8 x^{7} - 4 x^{3} + 5$

Step 1

Rewrite the equation.

$d u = (8 x^{7} - 4 x^{3} + 5) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d u = \int 8 x^{7} - 4 x^{3} + 5 d x$

Step 2.2

Apply the constant rule.

$u + C_{1} = \int 8 x^{7} - 4 x^{3} + 5 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$u + C_{1} = \int 8 x^{7} d x + \int - 4 x^{3} d x + \int 5 d x$

Step 2.3.2

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$u + C_{1} = 8 \int x^{7} d x + \int - 4 x^{3} d x + \int 5 d x$

Step 2.3.3

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$u + C_{1} = 8 (\frac{1}{8} x^{8} + C_{2}) + \int - 4 x^{3} d x + \int 5 d x$

Step 2.3.4

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

$u + C_{1} = 8 (\frac{1}{8} x^{8} + C_{2}) - 4 \int x^{3} d x + \int 5 d x$

Step 2.3.5

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

$u + C_{1} = 8 (\frac{1}{8} x^{8} + C_{2}) - 4 (\frac{1}{4} x^{4} + C_{3}) + \int 5 d x$

Step 2.3.6

Apply the constant rule.

$u + C_{1} = 8 (\frac{1}{8} x^{8} + C_{2}) - 4 (\frac{1}{4} x^{4} + C_{3}) + 5 x + C_{4}$

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 $\frac{1}{8}$ and $x^{8}$ .

$u + C_{1} = 8 (\frac{x^{8}}{8} + C_{2}) - 4 (\frac{1}{4} x^{4} + C_{3}) + 5 x + C_{4}$

Step 2.3.7.1.2

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

$u + C_{1} = 8 (\frac{x^{8}}{8} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) + 5 x + C_{4}$

Step 2.3.7.2

Simplify.

$u + C_{1} = x^{8} - x^{4} + 5 x + C_{5}$

Step 2.4

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

$u = x^{8} - x^{4} + 5 x + K$