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Which of the following can be converted to the form \int w^{10} dw using a substitution? a. \int...

Question:

Which of the following can be converted to the form {eq}\int w^{10} dw {/eq} using a substitution?

a. {eq}\int (x^3 +4)^{10} dx {/eq}

b. {eq}\int x(x^3 +4)^{10} dx {/eq}

c. {eq}\int x^2(x^3 +4)^{10} dx {/eq}

d. {eq}\int x^3(x^3 +4)^{10} dx {/eq}

e. {eq}\int x^4(x^3 +4)^{10} dx {/eq}

Integration by Substitution Method.

The substitution method can also be used in Leibniz?s method.

It is derived from the fundamental theorem of calculus.

The formula is:

{eq}\displaystyle\int x^n \ dx=\dfrac{x^{n+1}}{n+1}+c\\\\ {/eq}

Answer and Explanation:

Part A.)

{eq}\displaystyle\int (x^3+4)^{10}\ dx\\\\ {/eq}

Let:

{eq}x^3+4=z\\\\ 3x^2\ dx=dz\\\\ dx=\dfrac{dz}{3x^2}\\\\ {/eq}

or,

{eq}x^2=(z-4)^{\frac{2}{3}}\\\\ {/eq}

Hence:

{eq}dx=\dfrac{dz}{3(z-4)^{\frac{2}{3}}}\\\\ {/eq}

Therefore:

{eq}=\displaystyle\int z^{10}\cdot \dfrac{dz}{3(z-4)^{\frac{2}{3}}}\\\\ {/eq}

The given integrand is not calculated by the substitution method.

Hence this is not the correct substitution.

Part B.)

{eq}\displaystyle\int x(x^3+4)^{10}dx\\\\ {/eq}

Substituting:

{eq}x^3+4=w\\\\ 3x^2\ dx=dw\\\\ {/eq}

Or,

{eq}x=(w-4)^{\frac{1}{3}}\\\\ =\displaystyle\int w^{10}\dfrac{dw}{3(w-4)^{\frac{1}{3}}}\\\\ {/eq}

Hence this is not the correct substitution.

Part C.)

{eq}\displaystyle\int x^2(x^3+4)^{10}dx\\\\ {/eq}

Substituting:

{eq}x^3+4=w\\\\ 3x^2\ dx=dw\\\\ x^2\ dx=\dfrac{dw}{3}\\\\ {/eq}

Therefore:

{eq}=\displaystyle\int w^{10}\cdot \dfrac{dw}{3}\\\\ =\dfrac{1}{3}\displaystyle\int w^{10}dw\\\\ {/eq}

This is the correct substitution as asked in question.

Therefore integrating it we get the result.

{eq}=\dfrac{1}{3}\left [ \dfrac{w^{11}}{11} \right ]+c\\\\ =\dfrac{1}{33}\left ( x^3+4 \right )^{11}+c\\\\ {/eq}

Part D.)

{eq}\displaystyle\int x^3(x^3+4)^{10}dx\\\\ \displaystyle\int x^2\cdot x(x^3+4)^{10}dx\\\\ {/eq}

Substituting:

{eq}x^3+4=w\\\\ 3x^2\ dx=dw\\\\ x^2\ dx=\dfrac{dw}{3}\\\\ {/eq}

Therefore:

{eq}=\displaystyle\int w^{10}\cdot \dfrac{dw}{3}\cdot (w-4)^{\frac{1}{3}}\\\\ {/eq}

This is also not the correct substitution.

Part E.)

{eq}\displaystyle\int x^4(x^3+4)dx\\\\ =\displaystyle\int x^2\cdot x^2 (x^3+4)dx\\\\ {/eq}

Substituting:

{eq}x^3+4=z\\\\ 3x^2\ dx=dw\\\\ {/eq}

Therefore integration will be:

{eq}=\displaystyle\int w^{10}\cdot \dfrac{dw}{3}\cdot (w-4)^{\frac{2}{3}}\\\\ {/eq}

This is also not the correct substitution.


Learn more about this topic:

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Integration Problems in Calculus: Solutions & Examples

from AP Calculus AB & BC: Homework Help Resource

Chapter 13 / Lesson 13
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