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Chapter 4: Q. 48 (page 362)

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess-and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.
$鈭� x^{3} e^{3 x^{4} - 2} d x$

Short Answer

Expert verified

The solution of the integral is $\frac{1}{12} e^{3 x^{4} - 2} + C .$

Step by step solution

01

Step 1. Given Information.

The given integral is $鈭� x^{3} e^{3 x^{4} - 2} d x .$

02

Step 2. Solve.

By solving the integral we get,

$鈭� x^{3} e^{3 x^{4} - 2} d x Let u = 3 x^{4} - 2, d u = 12 x^{3} = 鈭� (\frac{1}{12}) e^{u} d u = \frac{1}{12} 鈭� e^{u} d u = \frac{1}{12} e^{u} + C Substitute back u = 3 x^{4} - 2 = \frac{1}{12} e^{3 x^{4} - 2} + C$

03

Step 3. Verification.

To verify the answer we differentiate $\frac{1}{12} e^{3 x^{4} - 2} + C$ it.

On differentiating we get,

$\frac{1}{12} e^{3 x^{4} - 2} + C = \frac{d}{d x} \frac{1}{12} e^{3 x^{4} - 2} + \frac{d}{d x} C = \frac{1}{12} \frac{d}{d x} e^{3 x^{4} - 2} + \frac{d}{d x} C = \frac{1}{12} (12 x^{3}) e^{3 x^{4} - 2} + 0 = x^{3} e^{3 x^{4} - 2}$

Hence proved.

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Most popular questions from this chapter

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{鈭� 蟺}^{蟺} 鈥� (1 + \sin 鈦� x) d x$

If ${鈭�}_{- 2}^{3} f (x) d x = 4, {鈭�}_{- 2}^{6} f (x) d x = 9, {鈭�}_{- 2}^{3} g (x) d x = 2$ , and ${鈭�}_{3}^{6} g (x) d x = 3$ , then find the values of each definite integral in Exercises $29 - 40$ . If there is not enough information, explain why.

${鈭�}_{- 2}^{- 2} x (f (x) + 3)^{2} d x$

Fill in each of the blanks:

(a) $鈭� x^{6} d x = + C .$

(b) is an antiderivative of role="math" localid="1648619282178" $x^{6} .$

(c) The derivative of is $x^{6} .$

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

${鈭�}_{2}^{4} (x^{2} + 1) d x$

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

${鈭�}_{0}^{4} ((2 x - 3)^{2} + 5) d x$

See all solutions

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