Problem to solve
Find the indefinite integral \[\int[-7x^5+2x^4-2]\cdot3^xdx\]
Solution
STEP 1
Take the integral: integral3^x (-7 x^5 + 2 x^4 - 2) dx
STEP 2
Expanding the integrand 3^x (-7 x^5 + 2 x^4 - 2) gives -7 3^x x^5 + 2 3^x x^4 - 2 3^x: = integral(-7 3^x x^5 + 2 3^x x^4 - 2 3^x) dx
STEP 3
Integrate the sum term by term and factor out constants: = -7 integral3^x x^5 dx + 2 integral3^x x^4 dx - 2 integral3^x dx
STEP 4
For the integrand 3^x x^5, integrate by parts, integral f dg = f g - integral g df, where f = x^5, dg = 3^x dx, df = 5 x^4 dx, g = 3^x/log(3): = -(7 3^x x^5)/log(3) + (2 + 35/log(3)) integral3^x x^4 dx - 2 integral3^x dx
STEP 5
For the integrand 3^x x^4, integrate by parts, integral f dg = f g - integral g df, where f = x^4, dg = 3^x dx, df = 4 x^3 dx, g = 3^x/log(3): = (35 3^x x^4)/(log^2(3)) + (2 3^x x^4)/log(3) - (7 3^x x^5)/log(3) + (-140/(log^2(3)) - 8/log(3)) integral3^x x^3 dx - 2 integral3^x dx
STEP 6
For the integrand 3^x x^3, integrate by parts, integral f dg = f g - integral g df, where f = x^3, dg = 3^x dx, df = 3 x^2 dx, g = 3^x/log(3): = -(140 3^x x^3)/(log^3(3)) - (8 3^x x^3)/(log^2(3)) + (35 3^x x^4)/(log^2(3)) + (2 3^x x^4)/log(3) - (7 3^x x^5)/log(3) + (420/(log^3(3)) + 24/(log^2(3))) integral3^x x^2 dx - 2 integral3^x dx
STEP 7
For the integrand 3^x x^2, integrate by parts, integral f dg = f g - integral g df, where f = x^2, dg = 3^x dx, df = 2 x dx, g = 3^x/log(3): = (140 3^(x + 1) x^2)/(log^4(3)) + (8 3^(x + 1) x^2)/(log^3(3)) - (140 3^x x^3)/(log^3(3)) - (8 3^x x^3)/(log^2(3)) + (35 3^x x^4)/(log^2(3)) + (2 3^x x^4)/log(3) - (7 3^x x^5)/log(3) + (-840/(log^4(3)) - 48/(log^3(3))) integral3^x x dx - 2 integral3^x dx
STEP 8
For the integrand 3^x x, integrate by parts, integral f dg = f g - integral g df, where f = x, dg = 3^x dx, df = dx, g = 3^x/log(3): = -(280 3^(x + 1) x)/(log^5(3)) - (16 3^(x + 1) x)/(log^4(3)) + (140 3^(x + 1) x^2)/(log^4(3)) + (8 3^(x + 1) x^2)/(log^3(3)) - (140 3^x x^3)/(log^3(3)) - (8 3^x x^3)/(log^2(3)) + (35 3^x x^4)/(log^2(3)) + (2 3^x x^4)/log(3) - (7 3^x x^5)/log(3) + (-2 + 840/(log^5(3)) + 48/(log^4(3))) integral3^x dx
STEP 9
The integral of 3^x is 3^x/log(3): = -(7 3^x x^5)/log(3) + (35 3^x x^4)/(log^2(3)) + (2 3^x x^4)/log(3) - (140 3^x x^3)/(log^3(3)) - (8 3^x x^3)/(log^2(3)) + (140 3^(x + 1) x^2)/(log^4(3)) + (8 3^(x + 1) x^2)/(log^3(3)) - (280 3^(x + 1) x)/(log^5(3)) - (16 3^(x + 1) x)/(log^4(3)) + (3^x (-2 + 840/(log^5(3)) + 48/(log^4(3))))/log(3) + constant
STEP 10
Which is equal to: Answer: | | = (3^x (-7 x^5 log^5(3) + x^4 log^4(3) (35 + log(9)) - 4 x^3 log^3(3) (35 + log(9)) + 12 x^2 log^2(3) (35 + log(9)) - 24 x log(3) (35 + log(9)) + 840 - 2 log^5(3) + 48 log(3)))/(log^6(3)) + constant
More problems
Number
001Find the indefinite integral \(\int\frac{x^2+2}{x^3+5x^2+2x-8} dx\)
002Find the indefinite integral \(\int\frac{4x^4-2}{x^3+4x^2+x-6} dx\)
003Find the indefinite integral \(\int\frac{4x^4-22x^3+11x^2-99x-34}{x^3-6x^2+4x-24} dx\)
004Find the indefinite integral \(\int\frac{3x^4+4x^2+5x}{(x-1)(x^2+1)^2} dx\)
005Find the indefinite integral \(\int\frac{x^6+6x^4+8x^2-4}{(x^2+4)(x^2+3)} dx\)
006Find the indefinite integral \(\int(-4x^4+6x^2+8x-1)\cos(x) dx\)
007Find the indefinite integral \(\int(-7x^5+2x^4-2)\cdot3^xdx\)
008Find the indefinite integral \(\int\arcsin^4(x) dx\)
009Find the indefinite integral \(\int\sin(\ln(x)) dx\)
010Find the indefinite integral \(\int(12x^{15}-3x^7-4x^3)\cdot e^{2x^4} dx\)
011Find the indefinite integral \(\int\frac{\sqrt{(3x^2+4)^3}}{x^4} dx\)
012Find the indefinite integral \(\int\frac{x^2}{\sqrt{2x^2-5}} dx\)
013Find the indefinite integral \(\int\frac{x^6}{\sqrt{(x^2+4)^3}} dx\)
014Find the indefinite integral \(\int x^2\sqrt{4-3x^2} dx\)
015Find the indefinite integral \(\int x^4\cdot\sqrt{x^2+4x+5} dx\)
016Find the indefinite integral \(\int\frac{\sqrt{x+2}}{x+1} dx\)
017Find the indefinite integral \(\int\frac{\sqrt{x+2}-4}{5-\sqrt{x+2}} dx\)
018Find the indefinite integral \(\int\frac{3\sqrt[3]{2x-1}-4}{\sqrt[6]{2x-1}-1} dx\)
019Find the indefinite integral \(\int\sqrt{\frac{x+3}{1-x}} dx\)
020Find the indefinite integral \(\int\frac{\sqrt[3]{x+1}-\sqrt{x+1}}{x} dx\)
021Find the indefinite integral \(\int\frac{\sin(x)-3\cos(x)}{\cos(x)-2} dx\)
022Find the indefinite integral \(\int\frac{3\sec(x)-\cot(x)}{1-\csc(x)} dx\)
023Find the indefinite integral \(\int\frac{\tan^2(x)+2}{\tan(x)+2} dx\)
024Find the indefinite integral \(\int\frac{3\sin^4(x)-2\cos^2(x)}{1+\sin^2(x)} dx\)
025Find the indefinite integral \(\int\tan^4(x)\sec^5(x) dx\)
Back
Edited by tododemate® - 2021