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Chapter 5: Q. 68 (page 452)

Solve each of the definite integrals in Exercises 67鈥�76.

鈭�03蟺sin5xdx

Short Answer

Expert verified

The solution of the integral is1615.

Step by step solution

01

Step 1. Given Information

The given integral is鈭�03蟺sin5xdx.

02

Step 2. Rewrite and substitute

  • Use the trigonometric identities to rewrite the given integral as follows:

鈭�03蟺sin5xdx=鈭�03蟺1-cos2x2sinxdx

  • Assume that cos(x)=u. So, -sin(x)dx=du
  • Substitute and simplify the obtained integral.

鈭�0蟺1-cos2x2sinxdx=-鈭�cos-10cos-13蟺1-u22du=鈭�1-1-1+2u2-u4du=鈭�-111-2u2+u4du

03

Step 3. Integrate

  • Integrate the obtained integral.

鈭�-111-2u2+u4du=u-23u3+u55-11=1-23+15+1-23+15=1615

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Most popular questions from this chapter

Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial x2-x+3 as the quotient above the top line, and the polynomial 3x 鈭� 1 at the bottom as the remainder. Thenp(x)=___andp(x)q(x)=____

Solve given definite integral.

鈭�12鈥�39鈭�2x23/2dx

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x鈭�3x2

Solve the integral:鈭�(x-ex)2dx

Why doesn鈥檛 the definite integral鈭�231-x2dx make sense? (Hint: Think about domains.)
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