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Chapter 4: Q. 50 (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. (Hint for Exercise 54: tanx=sinxcosx).

鈭�(sec2x+csc2x)dx.

Short Answer

Expert verified

The value of the integral istanx-cotx+c.

Step by step solution

01

Step 1. Given Information.

Given an integral鈭�(sec2x+csc2x)dx.

02

Step 2. Formula involved. 

鈭�sec2xdx=tanx,and鈭�csc2xdx=-cotx.

03

Step 3. Solving the integral. 

鈭�(sec2x+csc2x)dx=鈭�sec2xdx+鈭�csc2xdx.=tan(x)-cot(x)+c.

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Most popular questions from this chapter

Show by exhibiting a counterexample that, in general, 鈭�f(x)g(x)dx鈮�鈭�f(x)dx鈭�g(x)dx. In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

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�32鈥�x2+52xdx

Suppose f is positive on (鈭掆垶, 鈭�1] and [2,鈭�) and negative on the interval [鈭�1, 2]. Write (a) the signed area and (b) the absolute area between the graph of f and the x-axis on [鈭�3, 4] in terms of definite integrals that do not involve absolute values.

Show that F(x)=sinx-xcosx+2 is an antiderivative of f(x)=sinx.

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value.

limn鈫�鈭�鈭�k=1nk2+k+1n
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