What is the integral of (sec^2(5x)+(x+5)^2) ?

The integral of (sec^2(5x)+(x+5)^2) is 1/5 tan(5x)+(x^3)/3+5x^2+25x+C

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integral of (sec^2(5x)+(x+5)^2)

\int (sec^{2} (5 x) + (x + 5)^{2}) d x

\frac{1}{5} tan (5 x) + \frac{x ^{3}}{3} + 5 x^{2} + 25 x + C

Solution steps

\int sec^{2} (5 x) + (x + 5)^{2} 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 sec^{2} (5 x) d x + \int (x + 5)^{2} d x

\int sec^{2} (5 x) d x = \frac{1}{5} tan (5 x)

\int (x + 5)^{2} d x = \frac{x ^{3}}{3} + 5 x^{2} + 25 x

= \frac{1}{5} tan (5 x) + \frac{x ^{3}}{3} + 5 x^{2} + 25 x

Add a constant to the solution

= \frac{1}{5} tan (5 x) + \frac{x ^{3}}{3} + 5 x^{2} + 25 x + C

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What is the integral of (sec^2(5x)+(x+5)^2) ?
The integral of (sec^2(5x)+(x+5)^2) is 1/5 tan(5x)+(x^3)/3+5x^2+25x+C