Mister Exam

Other calculators

Solving integrals/ sqrt(1+3sin(x))cos(x)dx

Integral of sqrt(1+3sin(x))cos(x)dx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                           
  /                           
 |                            
 |    ______________          
 |  \/ 1 + 3*sin(x) *cos(x) dx
 |                            
/                             
0
∫013sin⁡(x)+1cos⁡(x) dx\int\limits_{0}^{1} \sqrt{3 \sin{\left(x \right)} + 1} \cos{\left(x \right)}\, dx 
Integral(sqrt(1 + 3*sin(x))*cos(x), (x, 0, 1))
Detail solution

  1. Let u=3sin⁡(x)+1u = 3 \sin{\left(x \right)} + 1.

    Then let du=3cos⁡(x)dxdu = 3 \cos{\left(x \right)} dx and substitute du3\frac{du}{3}:

    ∫u3 du\int \frac{\sqrt{u}}{3}\, du

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫u du=∫u du3\int \sqrt{u}\, du = \frac{\int \sqrt{u}\, du}{3}

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫u du=2u323\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}

      So, the result is: 2u329\frac{2 u^{\frac{3}{2}}}{9}

    Now substitute uu back in:

    2(3sin⁡(x)+1)329\frac{2 \left(3 \sin{\left(x \right)} + 1\right)^{\frac{3}{2}}}{9}

  2. Add the constant of integration:

    2(3sin⁡(x)+1)329+constant\frac{2 \left(3 \sin{\left(x \right)} + 1\right)^{\frac{3}{2}}}{9}+ \mathrm{constant}

The answer is:

2(3sin⁡(x)+1)329+constant\frac{2 \left(3 \sin{\left(x \right)} + 1\right)^{\frac{3}{2}}}{9}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                    
 |                                                  3/2
 |   ______________                 2*(1 + 3*sin(x))   
 | \/ 1 + 3*sin(x) *cos(x) dx = C + -------------------
 |                                           9         
/
∫3sin⁡(x)+1cos⁡(x) dx=C+2(3sin⁡(x)+1)329\int \sqrt{3 \sin{\left(x \right)} + 1} \cos{\left(x \right)}\, dx = C + \frac{2 \left(3 \sin{\left(x \right)} + 1\right)^{\frac{3}{2}}}{9}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
          ______________       ______________       
  2   2*\/ 1 + 3*sin(1)    2*\/ 1 + 3*sin(1) *sin(1)
- - + ------------------ + -------------------------
  9           9                        3
−29+21+3sin⁡(1)9+21+3sin⁡(1)sin⁡(1)3- \frac{2}{9} + \frac{2 \sqrt{1 + 3 \sin{\left(1 \right)}}}{9} + \frac{2 \sqrt{1 + 3 \sin{\left(1 \right)}} \sin{\left(1 \right)}}{3} 
=
= 
          ______________       ______________       
  2   2*\/ 1 + 3*sin(1)    2*\/ 1 + 3*sin(1) *sin(1)
- - + ------------------ + -------------------------
  9           9                        3
−29+21+3sin⁡(1)9+21+3sin⁡(1)sin⁡(1)3- \frac{2}{9} + \frac{2 \sqrt{1 + 3 \sin{\left(1 \right)}}}{9} + \frac{2 \sqrt{1 + 3 \sin{\left(1 \right)}} \sin{\left(1 \right)}}{3} 
-2/9 + 2*sqrt(1 + 3*sin(1))/9 + 2*sqrt(1 + 3*sin(1))*sin(1)/3
Numerical answer [src] 
1.24811743022586
1.24811743022586

    Use the examples entering the upper and lower limits of integration.

    © Mister Exam – Calculator