> with(student): PROBLEMA 281 > Int((3*x+1)^2,x)=int((3*x+1)^2,x); 1 ⌠ ( 3 x + 1 )2 dx = ( 3 x + 1 )3 ⌡ 9 > Int(3*x/sqrt(x^2+1),x)=int(3*x/sqrt(x^2+1),x); ⌠ x 3 dx = 3 x 2 + 1 2 x +1 ⌡ > Int(sin(x)*cos(x),x)=int(sin(x)*cos(x),x); 1 ⌠ sin( x ) cos( x ) dx = − cos( x )2 ⌡ 2 > Int(ln(x)/x,x)=int(ln(x)/x,x); ⌠ 1 ln( x ) dx = ln( x )2 2 x ⌡ > Int(1/(4+x^2),x)=int(1/(4+x^2),x); ⌠ 1 1 1 x d x = arctan 2 2 4 + x2 ⌡ > Int(1/x*ln(x),x)=int(1/x*ln(x),x); ⌠ 1 ln( x ) d = x ln( x )2 2 x ⌡ > Int((x+1)/(x^2-1),x)=int((x+1)/(x^2-1),x);Para nosotros ln(|x-1|). ⌠ x+1 dx = ln( x − 1 ) x2 − 1 ⌡ > simplify(Int(1/((x-1)^2+2),x)=int(1/((x-1)^2+2),x)); ⌠ 1 1 1 ( x − 1 ) 2 d = x 2 arctan 2 2 x2 − 2 x + 3 ⌡ > Int(exp(sqrt(x))/sqrt(x),x)=int(exp(sqrt(x))/sqrt(x),x); ⌠ ( x) e ( x) dx = 2 e x ⌡ > Int(arctan(x)/(1+x^2),x)=int(arctan(x)/(1+x^2),x); ⌠ arctan( x ) 1 dx = arctan( x )2 2 2 x +1 ⌡ > Int(x*sqrt(x^2+1),x)=int(x*sqrt(x^2+1),x); (3 / 2) ⌠ 1 2 2 x x + 1 dx = ( x + 1 ) 3 ⌡ > Int((x^3-1)/(x-1),x)=int((x^3-1)/(x-1),x); ⌠ 3 x − 1 1 3 1 2 d x = x + x +x 3 2 x−1 ⌡ > Int(sin(x)/cos(x)^2,x)=int(sin(x)/cos(x)^2,x); ⌠ sin( x ) 1 dx = 2 cos( x ) cos( x ) ⌡ > Int(exp(2*ln(x)),x)=int(exp(2*ln(x)),x); 1 ⌠ 2 x dx = x3 ⌡ 3 > Int(exp(x)/sqrt(1-exp(x)),x)=int(exp(x)/sqrt(1-exp(x)),x); ⌠ ex dx = −2 1 − ex ⌡ 1 − ex PROBLEMA 282 > restart:with(student): > Int(ln(x),x)=value(intparts(Int(ln(x),x),ln(x))); ⌠ ln( x ) dx = ln( x ) x − x ⌡ > Int(x^2*exp(x),x)=value(intparts(Int(x^2*exp(x),x),x)); ⌠ 2 x x e dx = x ( x ex − ex ) − x ex + 2 ex ⌡ > Int(x^2*ln(x),x)=value(intparts(Int(x^2*ln(x),x),x^2)); 2 8 ⌠ 2 x ln( x ) dx = x2 ( ln( x ) x − x ) − x3 ln( x ) + x3 ⌡ 3 9 > Int((x*arcsin(x))/sqrt(1-x^2),x)=value(intparts(Int((x*arcsin(x) )/sqrt(1-x^2),x),arcsin(x))); ⌠ x arcsin( x ) dx = −arcsin( x ) 2 1 − x ⌡ 1 − x2 + x > Int(arctan(x),x)=value(intparts(Int(arctan(x),x),arctan(x))); 1 ⌠ arctan( x ) dx = arctan( x ) x − ln( 1 + x2 ) ⌡ 2 > Int(arcsin(x),x)=value(intparts(Int(arcsin(x),x),arcsin(x))); ⌠ arcsin( x ) dx = arcsin( x ) x + 1 − x2 ⌡ > Int(ln(x)^2,x)=value(intparts(Int(ln(x)^2,x),ln(x)^2)); ⌠ ln( x )2 dx = ln( x )2 x − 2 ln( x ) x + 2 x ⌡ > Int(cos(ln(x)),x)=value(intparts(Int(cos(ln(x)),x),cos(ln(x)))); 1 1 ⌠ cos( ln( x ) ) dx = cos( ln( x ) ) x + sin( ln( x ) ) x ⌡ 2 2 > Int(x*sin(x),x)=value(intparts(Int(x*sin(x),x),x)); ⌠ x sin( x ) dx = −x cos( x ) + sin( x ) ⌡ > Int(exp(x)*cos(x),x)=value(intparts(Int(exp(x)*cos(x),x),cos(x)) ); 1 1 ⌠ x e cos( x ) dx = ex cos( x ) + sin( x ) ex ⌡ 2 2 > Int(x^2*cos(x),x)=value(intparts(Int(x^2*cos(x),x),x^2)); ⌠ 2 x cos( x ) dx = x2 sin( x ) − 2 sin( x ) + 2 x cos( x ) ⌡ > Int((x^2-1)*exp(2*x),x)=value(intparts(Int((x^2-1)*exp(2*x),x),( x^2-1))); ⌠ 1 1 (2 x) 1 (2 x) (2 x) (2 x) 2 − xe + e dx = ( − 1 + x 2 ) e ( −1 + x ) e 2 2 4 ⌡ PROBLEMA 284 > Int(sin(x)^2,x)=int(sin(x)^2,x); 1 1 ⌠ sin( x )2 dx = − cos( x ) sin( x ) + x ⌡ 2 2 > %=x/2-sin(2*x)/4; 1 1 1 1 ⌠ sin( x )2 dx = − cos( x ) sin( x ) + x = x − sin( 2 x ) 2 2 2 4 ⌡ > Int(sin(t)^2*cos(t)^2,t)=int(sin(t)^2*cos(t)^2,t); 1 1 1 ⌠ sin( t )2 cos( t )2 dt = − sin( t ) cos( t )3 + cos( t ) sin( t ) + t ⌡ 4 8 8 > Int(sin(x)^3,x)=int(sin(x)^3,x); 1 2 ⌠ sin( x )3 dx = − sin( x )2 cos( x ) − cos( x ) ⌡ 3 3 > Int(sin(t)^3*cos(t)^2,t)=int(sin(t)^3*cos(t)^2,t); 1 2 ⌠ sin( t )3 cos( t )2 dt = − sin( t )2 cos( t )3 − cos( t )3 ⌡ 5 15 > Int(cos(3*x)*cos(x)^3,x)=int(cos(3*x)*cos(x)^3,x); 1 1 3 3 ⌠ cos( 3 x ) cos( x )3 dx = sin( 6 x ) + x + sin( 2 x ) + sin( 4 x ) ⌡ 48 8 16 32 > Int(sin(x)^5*cos(x),x)=int(sin(x)^5*cos(x),x); 1 ⌠ sin( x )5 cos( x ) dx = sin( x )6 ⌡ 6 > Int(tan(x),x)=int(tan(x),x); ⌠ tan( x ) dx = −ln( cos( x ) ) ⌡ > Int(cos(x)^4,x)=int(cos(x)^4,x); 1 3 3 ⌠ cos( x )4 dx = sin( x ) cos( x )3 + cos( x ) sin( x ) + x ⌡ 4 8 8 > Int(sin(t)*cos(t)^3,t)=int(sin(t)*cos(t)^3,t); 1 ⌠ sin( t ) cos( t )3 dt = − cos( t )4 ⌡ 4 > Int(sin(x/2)*cos(x/3),x)=int(sin(x/2)*cos(x/3),x); ⌠ 1 3 5 1 1 sin x cos x dx = − cos x − 3 cos x 3 5 6 6 2 ⌡ PROBLEMA 286 > Int(exp(x)*cos(exp(x)),x)=int(exp(x)*cos(exp(x)),x); ⌠ x e cos( ex ) dx = sin( ex ) ⌡ > Int(sin(2*x)*cos(2*x)^3,x)=int(sin(2*x)*cos(2*x)^3,x); 1 ⌠ sin( 2 x ) cos( 2 x )3 dx = − cos( 2 x )4 ⌡ 8 > Int(1/sqrt(7+8*x^2),x)=int(1/sqrt(7+8*x^2),x); ⌠ 1 1 2 dx = 2 arcsinh 14 x 4 7 7 + 8 x2 ⌡ > Int(sin(ln(x))/x,x)=int(sin(ln(x))/x,x); ⌠ sin( ln( x ) ) dx = −cos( ln( x ) ) x ⌡ > Int(x*exp(-x^2),x)=int(x*exp(-x^2),x); ⌠ ( −x2 ) 1 ( −x2 ) x e dx = − e 2 ⌡ > Int(ln(x)/sqrt(x),x)=int(ln(x)/sqrt(x),x); ⌠ ln( x ) dx = 2 x ln( x ) − 4 x x ⌡ > Int(ln(x)^3/x,x)=int(ln(x)^3/x,x); ⌠ ln( x )3 1 dx = ln( x )4 4 x ⌡ > Int(tan(x)^2,x)=int(tan(x)^2,x); ⌠ tan( x )2 dx = tan( x ) − arctan( tan( x ) ) ⌡ > Int((sqrt(x)+ln(x))/x,x)=int((sqrt(x)+ln(x))/x,x); ⌠ x + ln( x ) 1 d = x x 2 + ln( x )2 x 2 ⌡ > Int(x^2/(x^2+2),x)=int(x^2/(x^2+2),x); ⌠ 2 x 1 x x − d = 2 arctan x 2 2 2 x + 2 ⌡ > Int(exp(2*x)/sqrt(exp(x)+1),x)=int(exp(2*x)/sqrt(exp(x)+1),x); ⌠ (2 x) e (3 / 2) 2 dx = ( e x + 1 ) − 2 ex + 1 x 3 e +1 ⌡ > Int(exp(sqrt(x)),x)=int(exp(sqrt(x)),x); ⌠ ( x) ( x) ( x) e dx = 2 e x −2e ⌡ > Int(arcsin(x)/x^2,x)=int(arcsin(x)/x^2,x); ⌠ arcsin( x ) arcsin( x ) 1 dx = − − arctanh 2 2 x x 1 x − ⌡ > Int(1/(x*sqrt(1-ln(x)^2)),x)=int(1/(x*sqrt(1-ln(x)^2)),x); ⌠ 1 dx = arcsin( ln( x ) ) x 1 − ln( x )2 ⌡ > Int(3^x*exp(x),x)=int(3^x*exp(x),x); x ( x ln( 3 ) ) e e ⌠ x x 3 e dx = ⌡ ln( 3 ) + 1 > Int(sqrt(arcsin(x))/sqrt((1-x^2)),x)=int(sqrt(arcsin(x))/sqrt((1 -x^2)),x); ⌠ arcsin( x ) 2 (3 / 2) dx = arcsin( x ) 3 1 − x2 ⌡ > Int(x/sqrt(x^2+1),x)=int(x/sqrt(x^2+1),x); ⌠ x dx = 1 + x2 ⌡ > Int(x/exp(x),x)=int(x/exp(x),x); 1 + x2 ⌠ x x 1 dx = − − ex ex ex ⌡ > Int(ln(ln(x))/x,x)=int(ln(ln(x))/x,x); ⌠ ln( ln( x ) ) dx = ln( x ) ln( ln( x ) ) − ln( x ) x ⌡ > Int(x^2/(1+25*x^6),x)=int(x^2/(1+25*x^6),x); ⌠ x2 1 arctan( 5 x3 ) dx = 6 15 1 + 25 x ⌡