integral of 2xsqrt(2x-1) — step-by-step solution

⚠ 2x is read as the product 2*x: for a power, write 2^x

⚠ 2xsqrt(2x-1) is read as 2x*sqrt(2x-1): use * for multiplication, use ^ for a power

1) Let's compute the integral:

\int(2 x \cdot \sqrt{2 x - 1})dx

For

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

, we use integration by parts:

\int(u1 \cdot v2')dx = u1 \cdot v2 - \int(u1' \cdot v2)dx

, with u1 =

x

and v2' =

\sqrt{2 x - 1}

Thus,

u1'

1

2) Let's integrate v2':

\int(\sqrt{2 x - 1})dx

For the math expression:

\sqrt{2 x - 1}

, we know the derivative formula:

\frac{d}{d x} (a^{n}) = n a^{(n - 1)}

, so we try differentiating with ^(m+1)

3) Let's perform the differentiation:

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

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

\frac{d}{d x} ({(2 x - 1)}^{\frac{3}{2}})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{(n - 1)}

with f =

2 x - 1

\frac{3}{2} \cdot \frac{d}{d x} (2 x - 1) \cdot {(2 x - 1)}^{(\frac{3}{2} - 1)}

ℹ

\frac{3}{2} \cdot \frac{d}{d x} (2 x - 1) \cdot {(2 x - 1)}^{(\frac{3}{2} - 1)}

\frac{3 \cdot \frac{d}{d x} (2 x - 1) \cdot {(2 x - 1)}^{(\frac{3}{2} - 1)}}{2}

‖

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

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

‖

\frac{3}{2} - 1

\frac{3 - 1 \cdot 2}{2}

\frac{3 \cdot (\frac{d}{d x} (2 x) + 0) \cdot {(2 x - 1)}^{(\frac{3 - 1 \cdot 2}{2})}}{2}

ℹ

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

2

‖

-1 \cdot 2

-2

\frac{3 \cdot 2 \cdot {(2 x - 1)}^{(\frac{3 - 2}{2})}}{2}

ℹ

3 \cdot 2

6

‖

3 - 2

1

\frac{6 \cdot \sqrt{2 x - 1}}{2}

3 \cdot \sqrt{2 x - 1}

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

\frac{1}{3} \cdot {(2 x - 1)}^{\frac{3}{2}}

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

v2

is equal to

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

4) Let's compute the integral

u1 \cdot v2 - \int(u1' \cdot v2)dx

2 \cdot (x \cdot \frac{{(2 x - 1)}^{\frac{3}{2}}}{3} - \int(\frac{{(2 x - 1)}^{\frac{3}{2}}}{3})dx)

We attempt to reverse an integration: starting from

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

and

\frac{d}{d x} (a^{n}) = n a^{(n - 1)}

, we differentiate using ^(m+1)

5) Let's differentiate:

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

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

\frac{d}{d x} ({(2 x - 1)}^{\frac{5}{2}})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{(n - 1)}

with f =

2 x - 1

\frac{5}{2} \cdot \frac{d}{d x} (2 x - 1) \cdot {(2 x - 1)}^{(\frac{5}{2} - 1)}

ℹ

\frac{5}{2} \cdot \frac{d}{d x} (2 x - 1) \cdot {(2 x - 1)}^{(\frac{5}{2} - 1)}

\frac{5 \cdot \frac{d}{d x} (2 x - 1) \cdot {(2 x - 1)}^{(\frac{5}{2} - 1)}}{2}

‖

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

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

‖

\frac{5}{2} - 1

\frac{5 - 1 \cdot 2}{2}

\frac{5 \cdot (\frac{d}{d x} (2 x) + 0) \cdot {(2 x - 1)}^{(\frac{5 - 1 \cdot 2}{2})}}{2}

ℹ

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

2

‖

-1 \cdot 2

-2

\frac{5 \cdot 2 \cdot {(2 x - 1)}^{(\frac{5 - 2}{2})}}{2}

ℹ

5 \cdot 2

10

‖

5 - 2

3

\frac{10 \cdot {(2 x - 1)}^{\frac{3}{2}}}{2}

5 \cdot {(2 x - 1)}^{\frac{3}{2}}

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

2 \cdot (\frac{x \cdot {(2 x - 1)}^{\frac{3}{2}}}{3} - \frac{1}{15} \cdot {(2 x - 1)}^{\frac{5}{2}})

ℹ

- \frac{1}{15} \cdot {(2 x - 1)}^{\frac{5}{2}}

- \frac{{(2 x - 1)}^{\frac{5}{2}}}{15}

‖

\frac{x \cdot {(2 x - 1)}^{\frac{3}{2}}}{3} - \frac{{(2 x - 1)}^{\frac{5}{2}}}{15}

\frac{5 \cdot x \cdot {(2 x - 1)}^{\frac{3}{2}} - {(2 x - 1)}^{\frac{5}{2}}}{15}

2 \cdot (\frac{5 \cdot x \cdot {(2 x - 1)}^{\frac{3}{2}} - {(2 x - 1)}^{\frac{5}{2}}}{15})

ℹ

2 \cdot \frac{5 x \cdot {(2 x - 1)}^{\frac{3}{2}} - {(2 x - 1)}^{\frac{5}{2}}}{15}

\frac{2 \cdot (5 x \cdot {(2 x - 1)}^{\frac{3}{2}} - {(2 x - 1)}^{\frac{5}{2}})}{15}

‖

\frac{2 \cdot (5 x \cdot {(2 x - 1)}^{\frac{3}{2}} - {(2 x - 1)}^{\frac{5}{2}})}{15}

\frac{2 \cdot {(2 x - 1)}^{\frac{3}{2}} \cdot (5 x - (2 x - 1))}{15}

\frac{2 \cdot {(2 x - 1)}^{\frac{3}{2}} \cdot (5 x - (2 x - 1))}{15}

\frac{2 \cdot {(2 x - 1)}^{\frac{3}{2}} \cdot (5 x - 2 x + 1)}{15}

\frac{2 \cdot {(2 x - 1)}^{\frac{3}{2}} \cdot (3 x + 1)}{15}

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

\frac{2 \cdot {(2 x - 1)}^{\frac{3}{2}} \cdot (3 x + 1)}{15} + C