Integration techniques
Mathematics, winter semester 2016/2017
November 29, 2016
Material from the handbook “Calculus Early Transcendentals” by J. Steward, 6th edition.
Recommended reading: Sections 5.5, 7.1.
Problems
´
1. Evaluate the integral by making the given substitution. (a) cos (3x) dx,
5
´
´
1
y = 3x; (b) x x2 + 4 dx, y = x2 + 4; (c) (1−6t)
4 dt, u = 1 − 6t;
´ √
´ x√
´
x
x
(d) x x + 2dx, y = x + 2; (e) e e + 1dx, y = e + 1; (f) lnxx dx,
y = ln x.
2. Evaluate the indefinite integral.
´
´
´
´
3
5
5.5
(a) x2 e−x dx; (b) (3x − 2) dx; (c) sin x cos xdx; (d) (5x − 2) (x + 1) dx;
´ ex
´
(e) √ex +1 dx; (f) cot (2x) dx.
3. Evaluate the integral using integraton
´ by parts with the indicated choices
of u ´= f and dv = g 0 (x) dx. (a) x1.3 ln xdx,
= ln x dv = x1.3 dx;
´ 2 u−x/3
1
x
(b) ex/3 dx, u = x dv = ex/3 dx; (c) x e
dx, u = x dv =
−x/3
xe ´ dx; remark: instead of using integraton by parts you may predict
that´ x2 e−x/3 dx = ax2 + bx + c e−x/3 + C for some constants a, b, c;
(d) x cos (5x) dx, u = x dv = cos (5x) dx.
4. Evaluate the indefinite integral.
´
´
´
´
2
(a) x´2 cos (5x) dx; (b)
´ (x − 5) ln xdx; (c)
´ x (sin x) dx; (d) arctan xdx,
hint: arctan xdx = arctan x´ · 1dx; (e) e sin xdx, hint: use integraton
by parts twice or predict that ex sin xdx = ex (a sin x + b cos x) for some
constants a and b.
5. First make a substitution
√and then use
´
´ by parts to evalu´ 3integration
t2
ate the integral.
(a)
cos
5xdx;
(b)
t
e
dt;
(c)
x ln (x + 5) dx; (d)
√
´√
x+1
x + 1e
dx.
1