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Answer :
To evaluate the definite integral [tex]\(\int_0^1 7 x^6\left(2-x^7\right) d x\)[/tex], we can use a substitution method to simplify the process. Here’s a step-by-step guide on how to solve this integral:
1. Identify the substitution:
Let's set [tex]\( u = x^7 \)[/tex]. Then, the derivative [tex]\( du = 7x^6 dx \)[/tex]. This substitution indicates that [tex]\(x^6 dx = \frac{1}{7} du\)[/tex].
2. Change the integration limits:
When [tex]\( x = 0 \)[/tex], [tex]\( u = 0^7 = 0 \)[/tex].
When [tex]\( x = 1 \)[/tex], [tex]\( u = 1^7 = 1 \)[/tex].
So, the limits for [tex]\( u \)[/tex] are from 0 to 1.
3. Substitute and integrate:
Substitute [tex]\( u = x^7 \)[/tex] into the integral:
[tex]\[
\int_0^1 7x^6 (2 - x^7) \, dx = \int_0^1 (2 - u) \, du
\][/tex]
4. Break down the integral:
The integral can be split into two simpler integrals:
[tex]\[
\int_0^1 (2 - u) \, du = \int_0^1 2 \, du - \int_0^1 u \, du
\][/tex]
5. Calculate each integral:
- The integral of a constant: [tex]\(\int_0^1 2 \, du = 2u \Big|_0^1 = 2(1) - 2(0) = 2\)[/tex].
- The integral of [tex]\( u \)[/tex]: [tex]\(\int_0^1 u \, du = \frac{u^2}{2} \Big|_0^1 = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2}\)[/tex].
6. Combine results:
Subtract the results of the two integrals:
[tex]\[
\int_0^1 2 \, du - \int_0^1 u \, du = 2 - \frac{1}{2} = \frac{3}{2}
\][/tex]
Therefore, the value of the definite integral [tex]\(\int_0^1 7 x^6\left(2-x^7\right) dx\)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
1. Identify the substitution:
Let's set [tex]\( u = x^7 \)[/tex]. Then, the derivative [tex]\( du = 7x^6 dx \)[/tex]. This substitution indicates that [tex]\(x^6 dx = \frac{1}{7} du\)[/tex].
2. Change the integration limits:
When [tex]\( x = 0 \)[/tex], [tex]\( u = 0^7 = 0 \)[/tex].
When [tex]\( x = 1 \)[/tex], [tex]\( u = 1^7 = 1 \)[/tex].
So, the limits for [tex]\( u \)[/tex] are from 0 to 1.
3. Substitute and integrate:
Substitute [tex]\( u = x^7 \)[/tex] into the integral:
[tex]\[
\int_0^1 7x^6 (2 - x^7) \, dx = \int_0^1 (2 - u) \, du
\][/tex]
4. Break down the integral:
The integral can be split into two simpler integrals:
[tex]\[
\int_0^1 (2 - u) \, du = \int_0^1 2 \, du - \int_0^1 u \, du
\][/tex]
5. Calculate each integral:
- The integral of a constant: [tex]\(\int_0^1 2 \, du = 2u \Big|_0^1 = 2(1) - 2(0) = 2\)[/tex].
- The integral of [tex]\( u \)[/tex]: [tex]\(\int_0^1 u \, du = \frac{u^2}{2} \Big|_0^1 = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2}\)[/tex].
6. Combine results:
Subtract the results of the two integrals:
[tex]\[
\int_0^1 2 \, du - \int_0^1 u \, du = 2 - \frac{1}{2} = \frac{3}{2}
\][/tex]
Therefore, the value of the definite integral [tex]\(\int_0^1 7 x^6\left(2-x^7\right) dx\)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
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