QQuestionMathematics
QuestionMathematics
"Find the following indefinite integral:
integral square root of x left parenthesis 5 x plus 3 right parenthesis squared d x
a. 50 over 7 x to the power of 7 over 2 end exponent plus 12 x to the power of 5 over 2 end exponent plus 6 x to the power of 3 over 2 end exponent plus c
b. 50 over 7 x to the power of 7 over 2 end exponent plus 6 x to the power of 3 over 2 end exponent plus c
c. 175 over 2 x to the power of 7 over 2 end exponent plus 75 x to the power of 5 over 2 end exponent plus 27 over 2 x to the power of 3 over 2 end exponent plus c
d. 175 over 2 x to the power of 7 over 2 end exponent plus 27 over 2 x to the power of 3 over 2 end exponent plus c"
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Answer
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Step 1:: Identify the function we need to integrate.
\int \sqrt{x}(5x+3)^2 dx
Step 2:: We can simplify the integrand by expanding $(5x+ 3)^1$ and then taking the square root of $x$ outside the parentheses.
\int (25x^2+30x+9) \sqrt{x} dx
Step 3:: Now, we can distribute the $\sqrt{x}$ to each term inside the parentheses.
\int (25x^{5/2}+30x^{3/2}+9x^{1/2}) dx
Step 4:: Integrate each term separately.
\int 25x^{5/2} dx + \int 30x^{3/2} dx + \int 9x^{1/2} dx
Step 5:: Use the power rule for integration, $\int x^n dx = \frac{x^{n+ 1}}{n+ 1} + C$, where $n$ is a constant and $C$ is the constant of integration.
25\left(\frac{x^{7/2}}{7/2}\right) + 30\left(\frac{x^{5/2}}{5/2}\right) + 9\left(\frac{x^{3/2}}{3/2}\right) + C
Step 6:: Simplify the expression.
\frac{50}{7}x^{7/2} + \frac{60}{5}x^{5/2} + 6x^{3/2} + C
Step 7:: Further simplify the expression by combining the terms with similar exponents.
\frac{50}{7}x^{7/2} + \frac{12}{5}x^{5/2} + 6x^{3/2} + C
Step 8:: The expression above is an antiderivative of the given function.
Final Answer
\boxed{\frac{50}{7}x^{7 / 2} + \frac{12}{5}x^{5 / 2} + 6x^{3 / 2} + C}
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