en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Calculus >

integral of x^7(x^4+1)^{10}

Solution

\int x^{7} (x^{4} + 1)^{10} d x

Solution

\frac{x ^{48}}{48} + \frac{5 x ^{44}}{22} + \frac{9 x ^{40}}{8} + \frac{10 x ^{36}}{3} + \frac{105 x ^{32}}{16} + 9 x^{28} + \frac{35 x ^{24}}{4} + 6 x^{20} + \frac{45 x ^{16}}{16} + \frac{5 x ^{12}}{6} + \frac{x ^{8}}{8} + C

Solution steps

\int x^{7} (x^{4} + 1)^{10} d x

Expand

x^{7} (x^{4} + 1)^{10} : x^{47} + 10 x^{43} + 45 x^{39} + 120 x^{35} + 210 x^{31} + 252 x^{27} + 210 x^{23} + 120 x^{19} + 45 x^{15} + 10 x^{11} + x^{7}

= \int x^{47} + 10 x^{43} + 45 x^{39} + 120 x^{35} + 210 x^{31} + 252 x^{27} + 210 x^{23} + 120 x^{19} + 45 x^{15} + 10 x^{11} + x^{7} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{47} d x + \int 10 x^{43} d x + \int 45 x^{39} d x + \int 120 x^{35} d x + \int 210 x^{31} d x + \int 252 x^{27} d x + \int 210 x^{23} d x + \int 120 x^{19} d x + \int 45 x^{15} d x + \int 10 x^{11} d x + \int x^{7} d x

\int x^{47} d x = \frac{x ^{48}}{48}

\int 10 x^{43} d x = \frac{5 x ^{44}}{22}

\int 45 x^{39} d x = \frac{9 x ^{40}}{8}

\int 120 x^{35} d x = \frac{10 x ^{36}}{3}

\int 210 x^{31} d x = \frac{105 x ^{32}}{16}

\int 252 x^{27} d x = 9 x^{28}

\int 210 x^{23} d x = \frac{35 x ^{24}}{4}

\int 120 x^{19} d x = 6 x^{20}

\int 45 x^{15} d x = \frac{45 x ^{16}}{16}

\int 10 x^{11} d x = \frac{5 x ^{12}}{6}

\int x^{7} d x = \frac{x ^{8}}{8}

= \frac{x ^{48}}{48} + \frac{5 x ^{44}}{22} + \frac{9 x ^{40}}{8} + \frac{10 x ^{36}}{3} + \frac{105 x ^{32}}{16} + 9 x^{28} + \frac{35 x ^{24}}{4} + 6 x^{20} + \frac{45 x ^{16}}{16} + \frac{5 x ^{12}}{6} + \frac{x ^{8}}{8}

Add a constant to the solution

= \frac{x ^{48}}{48} + \frac{5 x ^{44}}{22} + \frac{9 x ^{40}}{8} + \frac{10 x ^{36}}{3} + \frac{105 x ^{32}}{16} + 9 x^{28} + \frac{35 x ^{24}}{4} + 6 x^{20} + \frac{45 x ^{16}}{16} + \frac{5 x ^{12}}{6} + \frac{x ^{8}}{8} + C

Graph

Popular Examples

integral of (9^x-12e^x-5sin(x)+8)

\int (9^{x} - 12 e^{x} - 5 sin (x) + 8) d x

integral of 2/(sqrt(1+x^2))

\int \frac{2}{1 + x ^{2}} d x

integral of (sin(x)+2cos(x))^2

\int (sin (x) + 2 cos (x))^{2} d x

integral of (3x+1)(x-2)

\int (3 x + 1) (x - 2) d x

integral of csc^2(5x)cot(5x)

\int csc^{2} (5 x) cot (5 x) d x

Frequently Asked Questions (FAQ)

What is the integral of x^7(x^4+1)^{10} ?
The integral of x^7(x^4+1)^{10} is