BASIC INTEGRATION MATH PROBLEMS

Problem 1 :

∫12 cos(4θ)dθ

Solution:

=12 cos(4𝜃) d𝜃=12cos(4𝜃) d𝜃 ∴ cos a𝜃 d𝜃=sin a𝜃a+C=12sin(4𝜃)4+C=3 sin(4𝜃)+C

Problem 2 :

∫7 cos(5x) dx

Solution:

=7 cos(5x) dx=7cos(5x) dx=7sin(5x)5+C=7 sin(5x)5+C

Problem 3 :

4 sinx3dx

Solution:

=4 sinx3dxLet x3=tdx=3 dt=43 sin t dt=4×3sin t dt=12(-cos t)+C=-12 cosx3+C

Problem 4 :

∫4e-7x dx

Solution:

=4e-7x dx=4e-7x dx=4e-7x-7+C=-4e-7x7+C

Problem 5 :

9ex4 dx

Solution:

=9ex4 dx=9ex4 dxLet x4=tdx=4 dt=94et dt=9×4et dt=36et+C=36 ex4+C

Problem 6 :

∫-5cos Ï€x dx

Solution:

=-5 cos(𝜋x) dx=-5cos(𝜋x) dx∴cos a𝜃 d𝜃=sin a𝜃a+C=-5sin(𝜋x)𝜋 +C=-5sin(𝜋x)𝜋+C

Problem 7 :

∫-13e6t dt

Solution:

=-13e6t dt=-13e6t dt=-13e6t6+C=-13e6t6+C

II. Evaluate the following definite integrals.

Problem 8 :

415x2-8x+5dx

Solution:

=415x2-8x+5dx=415x2 dx-418x dx+415 dx=5x33-4x2+5x41=5343-13-442-12+5(4-1)=53(64-1)-4(16-1)+5(3)=53(63)-4(15)+15=105-60+15=60

Problem 9 :

91x32+2x+3dx

Solution:

=91x32+2x+3dx=91x32 dx+912x dx+913 dx=x525291+2x2291+(3x)91=2x525+x2+3x91=2(9)525+92+3(9)-2(1)525+12+3(1)=2(3)55+81+27-25+1+3=4865+108-25+4=10265-225=10045=200.8

Problem 10 :

94x+13xdx

Solution:

=94x+13xdxt=xdt=12xdxdx=2x dt=2t dt=32t+13t2t dt=322t2+23dt=23t332+23=23(27-8)+23=23(19)+23=383+23=403=13.33

Problem 11 :

415x3dx

Solution:

=415x3dx=415x-3 dx=541x-3 dx=-5x-2241=-5132-12=-5-1532=7532=2.344

Problem 12 :

2-1(1+3t)t2 dt

Solution:

=2-1(1+3t)t2 dt=2-1t2+3t3dt=t33+3t442-1=233+3(2)44-(-1)33+3(-1)44=83+12+13-34=32+144+4-912=574=14.25

Problem 13 :

1-22t2-12 dt

Solution:

=1-22t2-12 dt=1-24t4-4t2+1dt=4t55-4t33+t1-2=4(1)55-4(1)33+1-4(-2)55-4(-2)33-2=45-43+1--1285+323-2=715+25415=26115=17.4

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