1. I=∫ $\frac{\text{dx}}{1 + 2\sqrt{x}}$

I=$\int\frac{\text{dx}}{1 + 2\sqrt{x}}$= $\int\frac{2t\ dt}{1 + 2t}$= ∫(1- $\frac{1}{1 + 2t}$)dt=∫dt-$\frac{1}{2}\int$ $\frac{2dt}{1 + 2t}$

x=t² $\overset{\Rightarrow}{\ }$dx=2t dt

I=t- $\frac{1}{2}$ ln(1+2t) = $\sqrt{x}$ - $\frac{1}{2}$ ln (1+ 2$\sqrt{x}$)

2. I=∫ $\frac{3x\ dx}{5 + {2x}^{4}}$

I=$\int\frac{3x\ dx}{5 + 2x^{4}}$ =$\ \frac{3}{2\sqrt{2}}$ $\int\frac{\text{dt}}{5 + t^{2}}$ = $\frac{3}{2\sqrt{2}}\int\frac{\sqrt{5}\text{\ du}}{5 + 5u^{2}}$ = $\frac{3\sqrt{5}}{10\sqrt{2}}\int\frac{\text{du}}{1 + u^{2}}$

$\sqrt{2}x^{2}$=t$\overset{\Rightarrow}{\ }$2$\sqrt{2}$x dx=dt ; t=$\sqrt{5}$u$\overset{\Rightarrow}{\ }$ dt = $\sqrt{5}$du

I = $\frac{3}{2\sqrt{10}}$ arc tg(u) = $\frac{3}{2\sqrt{10}}$ arc tg ( $\frac{t}{\sqrt{5}}$ ) = $\frac{3}{2\sqrt{10}}$ arc tg ($\sqrt{\frac{2}{5}}$ x²

3. I =∫ $\frac{\cos\left( x \right)\text{dx}}{\sqrt{1 + 3sin(x)}}$

I = $\int\frac{\cos\left( x \right)\text{dx}}{\sqrt{1 + 3sin(x)}}$ = $\frac{1}{3}$ $\int\frac{\text{dt}}{\sqrt{t}}$ = $\frac{2}{3}\sqrt{t}$ = $\frac{2}{3}$ $\sqrt{1 + 3\sin{(x)}}$

1+3sin(x) = t $\overset{\Rightarrow}{\ }$ 3 cos (x) dx = dt

4. I = ∫2x dx

I = ∫2x dx = x²

5. I= ∫( x⁴+ 2 ) dx

I= ∫( x⁴+ 2 ) dx = ∫x⁴ dx+2 ∫dx = $\frac{1}{5}x^{5}$+ 2x

6. I= $\int\frac{\text{dx}}{x^{3}}$

I= $\int\frac{\text{dx}}{x^{3}}$ = - $\frac{1}{2x^{2}}$

Wzory rachunku całkowego:

1. ∫ cdx = x

2. ∫ x^a dx = $\frac{1}{a + 1}$ x^a + 1

3. ∫ $\frac{\text{dx}}{x^{2} + 1}$ = arc tg (x)

4. ∫ $\frac{f^{'}(x)}{f(x)}$ dx = ln [ f(x) ]

5. ∫ cf(x) dx = c ∫f ( x ) dx

6. ∫ [f (x) + g(x)] dx = ∫ f (x) dx + ∫g(x) dx