Calculus Examples

$\frac{d y}{d x} = 7 x^{6} - \sec (x) \tan (x)$

Step 1

Rewrite the equation.

$d y = (7 x^{6} - \sec (x) \tan (x)) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int 7 x^{6} - \sec (x) \tan (x) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 7 x^{6} - \sec (x) \tan (x) d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 7 x^{6} d x + \int - \sec (x) \tan (x) d x$

Step 2.3.2

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

$y + C_{1} = 7 \int x^{6} d x + \int - \sec (x) \tan (x) d x$

Step 2.3.3

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

$y + C_{1} = 7 (\frac{1}{7} x^{7} + C_{2}) + \int - \sec (x) \tan (x) d x$

Step 2.3.4

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

$y + C_{1} = 7 (\frac{1}{7} x^{7} + C_{2}) - \int \sec (x) \tan (x) d x$

Step 2.3.5

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$y + C_{1} = 7 (\frac{1}{7} x^{7} + C_{2}) - (\sec (x) + C_{3})$

Step 2.3.6

Simplify.

Tap for more steps...

Step 2.3.6.1

Simplify.

$y + C_{1} = 7 (\frac{1}{7}) x^{7} - \sec (x) + C_{4}$

Step 2.3.6.2

Simplify.

Tap for more steps...

Step 2.3.6.2.1

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

$y + C_{1} = \frac{7}{7} x^{7} - \sec (x) + C_{4}$

Step 2.3.6.2.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 2.3.6.2.2.1

Cancel the common factor.

$y + C_{1} = \frac{7}{7} x^{7} - \sec (x) + C_{4}$

Step 2.3.6.2.2.2

Rewrite the expression.

$y + C_{1} = 1 x^{7} - \sec (x) + C_{4}$

Step 2.3.6.2.3

Multiply $x^{7}$ by $1$ .

$y + C_{1} = x^{7} - \sec (x) + C_{4}$

Step 2.4

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

$y = x^{7} - \sec (x) + K$