What is the integral of e^{ln((2x^2+3x-4)^3)}(4x+3) ?

The integral of e^{ln((2x^2+3x-4)^3)}(4x+3) is 4x^8+24x^7+22x^6-90x^5-(399x^4)/4+180x^3+88x^2-192x+C

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integral of e^{ln((2x^2+3x-4)^3)}(4x+3)

\int e^{l n ((2 x^{2} + 3 x - 4)^{3})} (4 x + 3) d x

4 x^{8} + 24 x^{7} + 22 x^{6} - 90 x^{5} - \frac{399 x ^{4}}{4} + 180 x^{3} + 88 x^{2} - 192 x + C

Solution steps

\int e^{l n ((2 x^{2} + 3 x - 4)^{3})} (4 x + 3) d x

Expand

e^{l n ((2 x^{2} + 3 x - 4)^{3})} (4 x + 3) : 32 x^{7} + 168 x^{6} + 132 x^{5} - 450 x^{4} - 399 x^{3} + 540 x^{2} + 176 x - 192

= \int 32 x^{7} + 168 x^{6} + 132 x^{5} - 450 x^{4} - 399 x^{3} + 540 x^{2} + 176 x - 192 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 32 x^{7} d x + \int 168 x^{6} d x + \int 132 x^{5} d x - \int 450 x^{4} d x - \int 399 x^{3} d x + \int 540 x^{2} d x + \int 176 x d x - \int 192 d x

\int 32 x^{7} d x = 4 x^{8}

\int 168 x^{6} d x = 24 x^{7}

\int 132 x^{5} d x = 22 x^{6}

\int 450 x^{4} d x = 90 x^{5}

\int 399 x^{3} d x = \frac{399 x ^{4}}{4}

\int 540 x^{2} d x = 180 x^{3}

\int 176 x d x = 88 x^{2}

\int 192 d x = 192 x

= 4 x^{8} + 24 x^{7} + 22 x^{6} - 90 x^{5} - \frac{399 x ^{4}}{4} + 180 x^{3} + 88 x^{2} - 192 x

Add a constant to the solution

= 4 x^{8} + 24 x^{7} + 22 x^{6} - 90 x^{5} - \frac{399 x ^{4}}{4} + 180 x^{3} + 88 x^{2} - 192 x + C

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What is the integral of e^{ln((2x^2+3x-4)^3)}(4x+3) ?
The integral of e^{ln((2x^2+3x-4)^3)}(4x+3) is 4x^8+24x^7+22x^6-90x^5-(399x^4)/4+180x^3+88x^2-192x+C