What is the integral of (sqrt(x^2+1)+x)^4 ?

The integral of (sqrt(x^2+1)+x)^4 is (8x^5)/5+(8x^3)/3+x+8/5 (x^2+1)^{5/2}-4/3 (x^2+1)^{3/2}+C

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integral of (sqrt(x^2+1)+x)^4

\int (x^{2} + 1 + x)^{4} d x

\frac{8 x ^{5}}{5} + \frac{8 x ^{3}}{3} + x + \frac{8}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{4}{3} (x^{2} + 1)^{\frac{3}{2}} + C

Solution steps

\int (x^{2} + 1 + x)^{4} d x

Expand

(x^{2} + 1 + x)^{4} : 8 x^{4} + 4 x^{3} x^{2} + 1 + 8 x^{2} + 4 x (x^{2} + 1)^{\frac{3}{2}} + 1

= \int 8 x^{4} + 4 x^{3} x^{2} + 1 + 8 x^{2} + 4 x (x^{2} + 1)^{\frac{3}{2}} + 1 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 8 x^{4} d x + \int 4 x^{3} x^{2} + 1 d x + \int 8 x^{2} d x + \int 4 x (x^{2} + 1)^{\frac{3}{2}} d x + \int 1 d x

\int 8 x^{4} d x = \frac{8 x ^{5}}{5}

\int 4 x^{3} x^{2} + 1 d x = 2 (\frac{2}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}})

\int 8 x^{2} d x = \frac{8 x ^{3}}{3}

\int 4 x (x^{2} + 1)^{\frac{3}{2}} d x = \frac{4}{5} (x^{2} + 1)^{\frac{5}{2}}

\int 1 d x = x

= \frac{8 x ^{5}}{5} + 2 (\frac{2}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}}) + \frac{8 x ^{3}}{3} + \frac{4}{5} (x^{2} + 1)^{\frac{5}{2}} + x

Simplify

\frac{8 x ^{5}}{5} + 2 (\frac{2}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}}) + \frac{8 x ^{3}}{3} + \frac{4}{5} (x^{2} + 1)^{\frac{5}{2}} + x : \frac{8 x ^{5}}{5} + \frac{8 x ^{3}}{3} + x + \frac{8}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{4}{3} (x^{2} + 1)^{\frac{3}{2}}

= \frac{8 x ^{5}}{5} + \frac{8 x ^{3}}{3} + x + \frac{8}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{4}{3} (x^{2} + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{8 x ^{5}}{5} + \frac{8 x ^{3}}{3} + x + \frac{8}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{4}{3} (x^{2} + 1)^{\frac{3}{2}} + C

\int \frac{cos ( x )}{sin ( x ) ( 1 - sin ( x ) )} d x

\frac{d}{d t} (\frac{v ( t )}{t})

cos^{3} (sin (2 x^{4} - x))

y = x^{2} - 7, y = 9

\int e^{- in wt} d t

What is the integral of (sqrt(x^2+1)+x)^4 ?
The integral of (sqrt(x^2+1)+x)^4 is (8x^5)/5+(8x^3)/3+x+8/5 (x^2+1)^{5/2}-4/3 (x^2+1)^{3/2}+C