Integración por fórmula directa

1.- ∫▒〖3x〗^4 dx = 3∫▒x^4 dx =
= 3(x^5/5) + C =
= 3/5 x5 + C

2.- ∫▒1/x^3 dx = ∫▒x^(-3) dx =
= x^(-3+1)/(-3+1) + C =
= x^(-2)/(-2) + C =
= - 1/〖2x〗^2+ C

3.- ∫▒〖5u〗^(3⁄2) du =
= 5∫▒u^(3⁄2) du =
= 5(u^(3⁄2+1)/(3⁄2+1)) + C =
= 5 u^(5⁄2)/(5/2) + C =
= 2u^(5⁄2) + C

4.- ∫▒2/√(3&x) dx =
=∫▒2/x^(1⁄3) dx =
= ∫▒〖2x〗^(-1⁄3) dx =
= 2∫▒x^(-1⁄3) dx =
= 2(x^(-1⁄3+1)/(-1⁄3+1)) + C

5.- ∫▒〖〖6t〗^2 √(3&t)〗 dt =
= 6∫▒t^(7⁄3) dt =
= 6(t^(7⁄3+1)/(7/3+1)) + C6.- ∫▒(〖4x〗^3+x^2 ) dx =
= ∫▒〖4x〗^3 dx + ∫▒x^2 dx =
= 4(x^4/4) + x^3/3 +C =
=x4 + 1/3 x3 + C

7.- ∫▒y^3 (〖2y〗^2-3)dy =
= ∫▒(〖2y〗^5-3y^3 ) dy == 2∫▒y^5 dy - 3∫▒y^3 dy =
= 2(y^6/6)-3(y^4/4) + C =
= 1/3 y6 - 3/4 y4 + C

8.- ∫▒(3-2t+t^2 ) dt =
= ∫▒〖3 dt〗 - ∫▒〖2t dt〗 + ∫▒t^2 dt =
= 3∫▒dt - 2∫▒〖t dt〗+∫▒t^2 dt =
= 3t-2(t^2/2) + t^3/3 + C =
= 3t – t2+ 3t + C

9.- ∫▒(〖8x〗^4+4x^3-〖6x〗^2-4x+5) dx =
= ∫▒〖〖8x〗^4 dx〗+∫▒〖〖4x〗^3 dx-〗 ∫▒〖〖6x〗^2 dx〗-∫▒〖4x dx+ ∫▒〖5 dx〗〗 =
=8∫▒〖x^4 dx〗+4∫▒〖x^3 dx-〗 6∫▒〖x^2 dx〗-4∫▒〖x dx+5∫▒〖 dx〗〗 =
= 8(x^5/5)+4(x^4/4)-6(x^3/3)-4(x^2/2)+5x+ C =
= 8/5 x5 + x4 - 2x3 - 2x2 + 5x + C

10.- ∫▒√x (x+1)dx = ∫▒〖x^(1⁄2)(x+1) dx〗 =
= ∫▒〖(x^(3⁄2)+x^(1⁄2) ) dx〗 =
= ∫▒x^(3⁄2) dx + ∫▒x^(1⁄2) dx =
= x^(5⁄2)/(5/2) +x^(3⁄2)/(3/2) + C =
= 2/5 x^(5⁄2) + 2/3 x^(3⁄2) + C
11.-∫▒(x^(3⁄2)-x)dx =
= ∫▒〖x^(3⁄2) dx〗 - ∫▒〖x dx〗 =
= x^(5⁄2)/(5/2) - x^2/2 + C =
= 2/5 x^(5⁄2) - 1/2 x^2 + C

12.- ∫▒(2/x^3 +3/x^2 +5 ) dx =
= ∫▒2/x^3 dx + ∫▒3/x^2 dx ∫▒〖5 dx〗=
= ∫▒〖2x〗^(-3) dx + ∫▒〖3x〗^(-2) dx + ∫▒5dx =
= 2∫▒x^(-3) dx + 3∫▒x^(-2) dx + 5∫▒dx =
= 2(x^(-2)/(-2)) + 3(x^(-1)/(-1)) + 5x + C
= -1/x^2 -3/x + 5x...