Help me with my Integral Calculus question. Here is a photo

okeleyeoyindamola2006

Answered question

2022-08-05

karton

Expert2023-06-02Added 613 answers

To evaluate the integral

\int x \sqrt{5 x + 1} d x

, we can use the substitution method.
Let's substitute

u

for the expression

5 x + 1

. Then we can solve for

d x

u = 5 x + 1

To find

d x

, we differentiate both sides of the equation with respect to

x

\frac{d u}{d x} = 5

Solving for

d x

d x = \frac{1}{5} d u

Now, let's substitute

u

and

d x

in terms of the variable

u

into the integral:

\int x \sqrt{5 x + 1} d x = \int x \sqrt{u} (\frac{1}{5} d u)

Rearranging the terms:

\frac{1}{5} \int x \sqrt{u} d u

To simplify this integral, we can use integration by parts. Let's consider

u

and

d v

u = x ⟹ d u = d x

d v = \sqrt{u} d u

Taking the derivatives and integrating:

v = \frac{2}{3} u^{\frac{3}{2}} ⟹ d v = \sqrt{u} d u

Applying integration by parts:

\frac{1}{5} \int x \sqrt{u} d u = \frac{1}{5} (x \cdot \frac{2}{3} u^{\frac{3}{2}} - \int \frac{2}{3} u^{\frac{3}{2}} d x)

Simplifying further:

\frac{1}{5} (\frac{2}{3} x u^{\frac{3}{2}} - \frac{2}{3} \int u^{\frac{3}{2}} d x)

Integrating

\int u^{\frac{3}{2}} d x

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

Replacing

u

with

5 x + 1

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

Simplifying the expression further if necessary.
Therefore, the integral

\int x \sqrt{5 x + 1} d x

evaluates to

\frac{2}{15} x (5 x + 1)^{\frac{3}{2}} - \frac{4}{75} (5 x + 1)^{\frac{5}{2}} + C

, where

C

represents the constant of integration.

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