Question

integrate x ^ 5 * (1 - x ^ 3) ^ 10 dx from 0 to 1 Evaluate​

Answer 1

Answer:

∫

1+x

3

x

5

dx

Let 1+x

3

=t

2

⟹3x

2

dx=2tdt

⟹x

2

dx=

3

2

tdt

=x

2

dx=

3

2

tdt

∫

t

3

2

t(t

2

−1)

dt

=

3

2

∫(t

2

−1)tdt

=

3

2

(

3

(1+x

3

)

3/2

−(1+x

3

)

1/2

)

Answer 2

Concept:

Integration is a method of bringing disparate parts together to form a whole. We find a function whose differential is known in integral calculus.  The definite integral is defined as the limit and summation that we used to obtain the net area between a function and the x-axis in the previous section.

Given:

The expression $\int\limits^1_0 {x^5(1-x^3)^{10}} \, dx$.

Find:

The integral value of the expression.

Solution:

$\int\limits^1_0 {x^5(1-x^3)^{10}} \, dx \\take \quad x^3=t, x^2 = \frac{dt}{3} \\\int\limits^1_0 {x^3x^2(1-x^3)^{10}} \, dx \\\\\int\limits^1_0 {t(1-t)^{10}} \frac{dt}{3}$

$\frac{1}{3} \int\limits^1_0 {t(1-t)^{10}} dt \\\\\\\int\limits^1_0 {x^{m-1}(1-x)^{n-1}} \, dx = B(m,n) \\\frac{1}{3} \int\limits^1_0 {t(1-t)^{10}} dt = \frac{1}{3}B(2,11)\\\\=\frac{1}{3}*\frac{1!*11!}{12!} \\\\=\frac{1}{396}$

Hence, the integral value is $\frac{1}{396}$.