integral of 7arccos(x) — step-by-step solution

1) Let's integrate:

\int 7 \cdot \arccos (x) d x

⇥Take the constant factor

7

outside the integral. For

7 \cdot \arccos (x)

, we use integration by parts:

\int 7 \cdot u1 \cdot v2' d x = 7 \cdot (u1 \cdot v2 - \int u1' \cdot v2 d x)

, with u1 =

\arccos (x)

and v2' =

1

⇥1.1) Let's differentiate u1:

\frac{d}{d x} (\arccos (x))

⇥ⓘ The derivative of arccos(x) is -1/sqrt(1 - x^2):

\frac{d}{d x} (\arccos (x))

- \frac{1}{\sqrt{1 - x^{2}}}

⇥

- \frac{1}{\sqrt{1 - x^{2}}}

⇥1.2) Let's integrate v2':

\int 1 d x

⇥ⓘ The derivative of

x

1

, so

x

is an antiderivative of

1

\int 1 d x

x

⇥

x

⇥1.3) Let's calculate

u1' \cdot v2

- \frac{1}{\sqrt{1 - x^{2}}} \cdot x

⇥ⓘ Multiplying a fraction by a term multiplies its numerator:

- \frac{1}{\sqrt{1 - x^{2}}} \cdot x

- \frac{1 \cdot x}{\sqrt{1 - x^{2}}}

- \frac{x}{\sqrt{1 - x^{2}}}

⇥

- \frac{x}{\sqrt{1 - x^{2}}}

⇥1.4) Let's calculate

7 \cdot (u1 \cdot v2 - \int u1' \cdot v2 d x)

7 \cdot (\arccos (x) \cdot x - \int (- \frac{x}{\sqrt{1 - x^{2}}}) d x)

⇥For the math expression:

\frac{x}{\sqrt{1 - x^{2}}}

, since (a^n)' = n*a'*a^(n - 1), we try differentiating

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

⇥1.5) Let's differentiate:

\frac{d}{d x} ({(1 - x^{2})}^{- \frac{1}{2} + 1})

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators:

- \frac{1}{2} + 1

\frac{- 1 + 2}{2}

⇥

\frac{d}{d x} ({(1 - x^{2})}^{\frac{- 1 + 2}{2}})

⇥ⓘ The numerator and the denominator are computed before the division:

- 1 + 2

1

⇥

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

⇥We differentiate using:

\frac{d}{d x} (\sqrt{f}) = \frac{d}{d x} (f) \cdot \frac{1}{2 \sqrt{f}}

(where f =

1 - x^{2}

)

⇥ⓘ The derivative of sqrt(x) is 1/(2*sqrt(x)):

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

\frac{d}{d x} (1 - x^{2}) \cdot \frac{1}{2 \cdot \sqrt{1 - x^{2}}}

⇥

\frac{d}{d x} (1 - x^{2}) \cdot \frac{1}{2 \cdot \sqrt{1 - x^{2}}}

⇥ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (1 - x^{2}) \cdot \frac{1}{2 \cdot \sqrt{1 - x^{2}}}

\frac{\frac{d}{d x} (1 - x^{2}) \cdot 1}{2 \cdot \sqrt{1 - x^{2}}}

\frac{\frac{d}{d x} (1 - x^{2})}{2 \cdot \sqrt{1 - x^{2}}}

■The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (1 - x^{2})

0 + \frac{d}{d x} (- x^{2})

⇥

\frac{0 + \frac{d}{d x} (- x^{2})}{2 \cdot \sqrt{1 - x^{2}}}

⇥ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (- x^{2})

\frac{d}{d x} (- x^{2})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (- x^{2})

- 2 x

⇥

- \frac{2 x}{2 \cdot \sqrt{1 - x^{2}}}

⇥ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

- \frac{2 x}{2 \cdot \sqrt{1 - x^{2}}}

- \frac{x}{\sqrt{1 - x^{2}}}

⇥

- \frac{x}{\sqrt{1 - x^{2}}}

⇥You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

⇥ⓘ The derivative of

- \sqrt{1 - x^{2}}

\frac{x}{\sqrt{1 - x^{2}}}

, so

- \sqrt{1 - x^{2}}

is an antiderivative of

\frac{x}{\sqrt{1 - x^{2}}}

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

- \sqrt{1 - x^{2}}

⇥

7 \cdot \arccos (x) \cdot x + 7 \cdot (- \sqrt{1 - x^{2}})

⇥ⓘ The minus sign is placed in front of the product:

7 \cdot (- \sqrt{1 - x^{2}})

- 7 \cdot \sqrt{1 - x^{2}}

⇥

7 \cdot \arccos (x) \cdot x - 7 \cdot \sqrt{1 - x^{2}}

Then you can continue the integration:

ⅰ

\int 7 \cdot \arccos (x) d x

7 \cdot \arccos (x) \cdot x - 7 \cdot \sqrt{1 - x^{2}}

7 \cdot \arccos (x) \cdot x - 7 \cdot \sqrt{1 - x^{2}}

We factor out

7

, the greatest common divisor of the coefficients:

7 \cdot \arccos (x) \cdot x - 7 \cdot \sqrt{1 - x^{2}}

7 \cdot (\arccos (x) \cdot x - \sqrt{1 - x^{2}})

Factor found:

x \cdot \arccos (x) - \sqrt{1 - x^{2}}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

7 \cdot (x \cdot \arccos (x) - \sqrt{1 - x^{2}}) + C