What is the integral of (e^x)/((25-e^{2x))^{3/2}} ?

The integral of (e^x)/((25-e^{2x))^{3/2}} is (e^x)/(25(25-e^{2x))^{1/2}}+C

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integral of (e^x)/((25-e^{2x))^{3/2}}

\int \frac{e ^{x}}{( 25 - e ^{2 x} ) ^{\frac{3}{2}}} d x

\frac{e ^{x}}{25 ( 25 - e ^{2 x} ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{e ^{x}}{( 25 - e ^{2 x} ) ^{\frac{3}{2}}} d x

Apply u-substitution

: \int \frac{1}{( 25 - u ^{2} ) ^{\frac{3}{2}}} d u

= \int \frac{1}{( 25 - u ^{2} ) ^{\frac{3}{2}}} d u

Apply Trigonometric Substitution

: \int \frac{1}{25 cos ^{2} ( v )} d v

= \int \frac{1}{25 cos ^{2} ( v )} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{25} \cdot \int \frac{1}{cos ^{2} ( v )} d v

Use the common integral:

\int \frac{1}{cos ^{2} ( v )} d v = tan (v)

= \frac{1}{25} tan (v)

Substitute back

= \frac{1}{25} tan (arcsin (\frac{1}{5} e^{x}))

Simplify

\frac{1}{25} tan (arcsin (\frac{1}{5} e^{x})) : \frac{e ^{x}}{25 ( 25 - e ^{2 x} ) ^{\frac{1}{2}}}

= \frac{e ^{x}}{25 ( 25 - e ^{2 x} ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{e ^{x}}{25 ( 25 - e ^{2 x} ) ^{\frac{1}{2}}} + C

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What is the integral of (e^x)/((25-e^{2x))^{3/2}} ?
The integral of (e^x)/((25-e^{2x))^{3/2}} is (e^x)/(25(25-e^{2x))^{1/2}}+C