Tutorial Exercise Find the general indefinite integral. 2V ( v 2 + 7 ) 2 dv Step 1 We'll begin by expanding ( v2 + 7)2 = 1 4 | 4 + 14 0 3 14 2 + 49 0 49 . Step 2 Therefore, 2v(v2 + 7)2 = 2v v . Submit Skip (you cannot come back) Tutorial Exercise Evaluate the integral. 1 . x (5 /x + 4 V/x ) ox Part 1 of 3 To find an antiderivative of f(x) = x(5Vx + 4\\x), we'll first convert the radicals to fractional powers and then distribute the X. (x) = x 5x 173 2 1/3 + 4x1/4 2 1/4 = 5x4/3 P 4/3 + 415/4 1 5/4 Part 2 of 3 An antiderivative of f(x) = 5x4/3 + 4x5/4 is F( x ) = x7/3 Submit Skip (you cannot come back) Tutorial Exercise Evaluate the indefinite integral . " v10 + ex dx Step 1 We must decide what to choose for u. If u = f(x), then du = f '(x) dx, and so it is helpful to look for some expression in ev 10 + ex dx for which the derivative is also present. We see that 10 + ex is part of this integral, and the derivative of 10 + et is et [ which is also present. Step 2 If we choose u = 10 + ex, then du = ex dx. If u = 10 + ex is substituted into ex v 10 + ex dx, then we have vio tedx = ex Judx = / vu(edx ) We must also convert ed dx into an expression involving u, but we already know that edx = 1 1 du . Step 3 Now , if u = 10 + ex, then ev 10 + ex ax = / Vudu= / 43/2 du. This evaluates as / 41/2 du = Submit Skip (you cannot come back) EXAMPLE 6 Evaluate the following integral. tan ( x ) dx SOLUTION First we write tangent in terms of sine and cosine: tan ( x ) dx = sin X_dx. cos( x) This suggests that we should substitute u = cos(x), since then du = dx and so: tan (x) dx =/ sin (x) dx cos(x) = - = -In(lul) + c - C (in terms of x)