Help to Integral Calculus Solutions & Examples

Use a table of integrals to evaluate the following indefinite integrals.
${\displaystyle \int \sin 3x\cos 2xdx}$

Solve the integral by substitution.
${\displaystyle \int \frac{{\sec }^2\left(\arctan x\right)}{1+x^2}dx}$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
${\displaystyle {\int }_{-\mathrm{\infty }}^0\frac{1}{3-4x}dx}$

Solve the integral.
${\displaystyle \int 3\sec (2x-1)\tan (2x-1)dx}$

Evaluate the integral.
${\displaystyle \int 3x^2+4x+\frac{4}{x^3}+xdx}$

Find the indefinite integral.
${\displaystyle \int \left(\sec y\right)(\tan y-\sec y)dy}$

Evaluate the integral.
${\displaystyle \int \frac{xdx}{\sqrt{1+x^2}}}$

Evaluate the integral.
${\displaystyle {\int }_{\frac{\pi }{6}}^{\frac{\pi }{2}}\frac{\sin 2x}{2\sin x}}$

Evaluate the integral or state that it is divergent.
${\displaystyle {\int }_{\frac{1}{2}}^2\frac{dx}{x\ln x}}$

Which one is the correct transformation?
Rewrite the function below from the given standard form into the transformational form of ${\displaystyle f\left(x\right)=q\left(x\right)+\frac{r\left(x\right)}{d\left(x\right)}}$ in order to identify the transformations.
Given Standard form: ${\displaystyle f\left(x\right)=\frac{5x-9}{x-3}}$
Dilation by a factor of 6 with respect to the x-axis, Translation right 3 units and up 5 units
Dilation by a factor of 6 with respect to the x-axis, Translation right 5 units and down 3 units
Dilation by a factor of 1/6 with respect to the y-axis, Translation right 5 units and up 3 units
Dilation by a factor of 1/6 with respect to the y-axis, Translation right 3 units and up 5 units
Dilation by a factor of 5 with respect to the x-axis, Translation left 3 units and up 6 units

Use integration by parts to evaluate the integral. Find u, du, dv and v. Find the antiderivative using integration by parts then evaluate antiderivative.
${\displaystyle {\int }_1^4\left(\ln x\right)xdx}$

Find the indefinite integral.
${\displaystyle \int \frac{x{e}^{2x}}{{(2x+1)}^2}dx}$

Evaluate the integral.
${\displaystyle \int {\cos }^{2}x\sin 2xdx}$

Evaluate the integral.
${\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}15{\sin }^{4}3x\cos 3xdx}$

Apply the transformation M to the polygon with the given vertices. Identify and describe the transformation. ${\displaystyle M:(x,y)\to (–x,–y)}$

Give a counterexample to show that the given transformation is not a linear transformation.
$T\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}xy\\ x+y\end{array}\right]T$

Evaluate the integral.
${\displaystyle \int t{e}^{-4t}dt}$

Evaluate the integrals.
${\displaystyle \int \frac{\cos \left(\sqrt{x}\right)}{\sqrt{x}}dx}$

Evaluate the integral.
${\displaystyle \int (10x^{\frac{1}{5}}+3e^x)dx}$

Evaluate the integral.
${\displaystyle \int {\cos }^{2}3xdx}$