Question

Решить интегралы. 3 примера

Answer 1

$1)int sin ^2left(xright)dx=int (frac{1-cos left(2xright)}{2})dx=frac{1}{2}cdot int :(1-cos left(2xright) )dx= \ =frac{1}{2}left(int :1dx-int cos left(2xright)dxright)=frac{1}{2}left(x-frac{1}{2}sin left(2xright)right)+C$
$2) int sin3(x)*cos5(x) dx = int( frac{sin left(5x+3xright)-sin left(5x-3xright)}{2})dx= \ frac{1}{2}cdot int :sin (left(5x+3xright)-sin left(5x-3xright) )dx= \ frac{1}{2}left(int (sin left(5x+3xright) )dx-int (sin left(5x-3xright)dxright)= \ =frac{1}{2}left(-frac{1}{8}cos left(8xright)-left(-frac{1}{2}cos left(2xright)right)right)= \ =frac{1}{2}left(-frac{1}{8}cos left(8xright)+frac{1}{2}cos left(2xright)right)+ C$
$3)ctg^2(x)dx = int ( frac{1}{sin^2 (x)}-1 ) dx = - ctg^2x-x+C$