Find the following indefinite integral:

integral left parenthesis 2 x plus 3 right parenthesis left parenthesis x squared plus 3 x minus 9 right parenthesis to the power of 15 d x


a.)
1 over 15 left parenthesis x squared plus 3 x right parenthesis left parenthesis x squared plus 3 x minus 9 right parenthesis to the power of 16 plus c


b.)
1 over 16 u to the power of 16 plus c


c.)
1 over 15 left parenthesis x squared plus 3 x minus 9 right parenthesis to the power of 16 plus c


d.)
1 over 16 left parenthesis x squared plus 3 x minus 9 right parenthesis to the power of 16 plus c

To solve the integral 

\[ \int (2x + 3)(x^2 + 3x - 9)^{15} \, dx, \]

we can use the substitution method. 

Let \( u = x^2 + 3x - 9 \). Then, we need to find \( du \).

\[ \frac{du}{dx} = 2x + 3. \]

So,

\[ du = (2x + 3) \, dx. \]

This means \( (2x + 3) \, dx = du \).

Now we can rewrite the integral in terms of \( u \):

\[ \int (2x + 3)(x^2 + 3x - 9)^{15} \, dx = \int u^{15} \, du. \]

The integral of \( u^{15} \) with respect to \( u \) is:

\[ \int u^{15} \, du = \frac{u^{16}}{16} + C. \]

Substituting back \( u = x^2 + 3x - 9 \),

\[ \frac{(x^2 + 3x - 9)^{16}}{16} + C. \]

Thus, the answer is

\[ \frac{(x^2 + 3x - 9)^{16}}{16} + C. \]

This corresponds to option

d.) \( \frac{1}{16} (x^2 + 3x - 9)^{16} + C \).

Would you like further details or have any questions? Here are 5 questions you might want to ask next:

1. How do you find the derivative of a composite function?
2. Can you explain the substitution method in integration?
3. What are the steps to solve a definite integral using substitution?
4. How do you verify the result of an indefinite integral?
5. Can you provide more examples of integration by substitution?

Tip: When using substitution in integration, ensure that the differential \( du \) matches the differential part of the original integral.

Learn how to solve the indefinite integral ∫(2x + 3)(x^2 + 3x - 9)^15 dx using the substitution method. This problem involves advanced integration techniques and the application of substitution principles. Suitable for advanced high school or college students.

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