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Chapter 11: Problem 12

Find the indefinite integral and check your result by differentiation. $$ \int 3 t^{4} d t $$

Short Answer

Expert verified
The integral of \(3t^4\) is \(\frac{3}{5} t^5 + C\).

Step by step solution

01

Apply Power Rule for Integration

Start by applying the power rule for integration. The function is \(3t^4\), where the variable \(t\) has a power of 4 and it is multiplied by a constant 3. According to power rule, the integral of \(t^n\) is \(\frac{1}{n+1} t^{n+1}\). Hence, the integral of \(3t^4\) would be \(3 \times \frac{1}{4 + 1} t^{4 + 1}\) which simplifies to \(\frac{3}{5} t^5\).
02

Include the Constant of Integration

It's important to remember that when finding an indefinite integral, there is an arbitrary constant of integration that should be added to the result. This is because the derivative of a constant is zero, so when differentiating, the constant 'disappears'. Essentially, any definite value could have been added to the original function, and it would still have the same derivative. Hence, the complete integral is \(\frac{3}{5} t^5 + C\), where \(C\) represents the constant of integration.
03

Validate by Differentiation

To confirm that the integral found is correct, differentiate the result and it should equate to the original function, \(3t^4\). The derivative of \(\frac{3}{5} t^5 + C\) is \(3t^4\), hence validating that the integral found is correct.

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Most popular questions from this chapter

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=x^{2}+4 x \quad[0,4] $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=-2 x+3, \quad[0,1] $$

The total cost of purchasing and maintaining a piece of equipment for \(x\) years can be modeled by \(C=5000\left(25+3 \int_{0}^{x} t^{1 / 4} d t\right)\) Find the total cost after (a) 1 year, (b) 5 years, and (c) 10 years.

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{0}^{1} x^{3}\left(x^{3}+1\right)^{3} d x $$

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{1 / 2}^{1}(x+1) \sqrt{1-x} d x $$
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