What is the integral of ln(at+sqrt(1+(at)^2)) ?

The integral of ln(at+sqrt(1+(at)^2)) is tln(sqrt(a^2t^2+1)+at)-(sqrt(a^2t^2+1))/a+C

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integral of ln(at+sqrt(1+(at)^2))

\int ln (a t + 1 + (a t)^{2}) d t

t ln (a^{2} t^{2} + 1 + a t) - \frac{a ^{2} t ^{2} + 1}{a} + C

Solution steps

\int ln (a t + 1 + (a t)^{2}) d t

Apply Integration By Parts

: t ln (a^{2} t^{2} + 1 + a t) - \int \frac{a t}{a ^{2} t ^{2} + 1} d t

= t ln (a^{2} t^{2} + 1 + a t) - \int \frac{a t}{a ^{2} t ^{2} + 1} d t

\int \frac{a t}{a ^{2} t ^{2} + 1} d t = \frac{a ^{2} t ^{2} + 1}{a}

= t ln (a^{2} t^{2} + 1 + a t) - \frac{a ^{2} t ^{2} + 1}{a}

Add a constant to the solution

= t ln (a^{2} t^{2} + 1 + a t) - \frac{a ^{2} t ^{2} + 1}{a} + C

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

\int \frac{1}{x ^{2} - 9 x + 18} d x

\int \frac{25 - x ^{2}}{x ^{2}} d x

y = \frac{x ^{10} - 5 x ^{6} - 12 x}{x ^{6}}

\frac{d y}{d x} - \frac{9}{x} y = \frac{y ^{3}}{x ^{13}}, y (1) = 1

What is the integral of ln(at+sqrt(1+(at)^2)) ?
The integral of ln(at+sqrt(1+(at)^2)) is tln(sqrt(a^2t^2+1)+at)-(sqrt(a^2t^2+1))/a+C