13.07 Further antidifferentiation
Find the following indefinite integrals, using C as the constant of integration:
\int \left(x - 5\right)^{4} dx
\int \dfrac{1}{\left(x + 6\right)^{3}} dx
\int 6 \left(x + 2\right)^{2} dx
\int \dfrac{4}{\left(x - 5\right)^{3}} dx
\int \left( 2 x + 3\right)^{3} dx
\int \left( 4 x - 3\right)^{ - 4 } dx
\int \dfrac{1}{\left( 2 x - 3\right)^{4}} dx
\int \dfrac{1}{\left( 4 x + 3\right)^{2}} dx
\int \left( 2 x - 5\right)^{\frac{1}{2}} dx
\int \left( 4 x + 5\right)^{ - \frac{1}{2} } dx
\int \left( 4 x + 5\right)^{\frac{5}{4}} dx
\int \left(1 - x\right)^{6} dx
\int 4 \left(5 - x\right)^{3} dx
\int \left(4 - 3 x\right)^{6} dx
\int 4 \left( 2 x + 5\right)^{3} dx
\int 4 \left(2 - 3 x\right)^{3} dx
\int \left(4 - 3 x\right)^{ - 5 } dx
\int \dfrac{5}{\left(5 - 4 x\right)^{3}} dx
\int \sqrt{7 - 4 x} \ dx
\int 3 \sqrt{ 2 x + 3} \ dx
\int \dfrac{5}{\sqrt{1 - 4 x}} dx
\int 16 \sqrt[3]{ 6 x + 5} \ dx
\int \left(5 + \left( 4 x + 1\right)^{3}\right) dx
\int \dfrac{5}{\sqrt{x - 2}} dx
\int \left(\sqrt{x - 2} + \sqrt{x + 3}\right) dx
Find the equation of y in terms of x, given that \dfrac{d y}{d x} = 12 \left(2x - 1\right)^{3} and y = 1 when x = 1.
Find the equation of the curve that has a gradient function \dfrac{d y}{d x} =5 \left(5x - 6\right)^{2} and the point \left(1,1 \right) lies on the curve.
For each of the following gradient functions, find the equation of y:
Consider the function \left(x^{2} - 3\right)^{5}.
Calculate \dfrac{d}{dx} \left(x^{2} - 3\right)^{5}.
Hence find \int 30 x \left(x^{2} - 3\right)^{4} dx.
Consider the function \left( 5 x^{2} + 10 x - 3\right)^{5}.
Calculate \dfrac{d}{dx} \left( 5 x^{2} + 10 x - 3\right)^{5}.
Find \int 150 \left(x + 1\right) \left( 5 x^{2} + 10 x - 3\right)^{4} dx.
Consider the function \left(x^{2} - 3 x + 6\right)^{5}.
Calculate \dfrac{d}{dx} \left(x^{2} - 3 x + 6\right)^{5}.
Hence find \int \left(3 - 2 x\right) \left(x^{2} - 3 x + 6\right)^{4} dx.
Consider the function \sqrt{ 4 x + 11}.
Calculate \dfrac{d}{dx} \left(\sqrt{ 4 x + 11}\right)
Hence find \int \dfrac{12}{\sqrt{ 4 x + 11}} dx.
Consider the function \dfrac{1}{\left(x^{2} + 7\right)^{3}}.
Calculate \dfrac{d}{dx} \left(\dfrac{1}{\left(x^{2} + 7\right)^{3}}\right).
Hence find \int - \dfrac{24 x}{\left(x^{2} + 7\right)^{4}} dx.
Consider the function y = e^{x^{3}}.
Calculate \dfrac{d y}{d x}.
Hence find \int 3 x^{2} e^{x^{3}} dx.
Consider the function y = e^{ 3 x^{4} - 5} and hence find 36 \int x^{3} e^{ 3 x^{4} - 5} dx.
Consider the function y = \sin \left(x^{5}\right) and hence find \int x^{4} \cos \left(x^{5}\right) dx.
Consider the function y = \cos \left(x^{6} + 3 x^{5}\right) and hence find 15 \int \left( 2 x^{5} + 5 x^{4}\right) \sin \left(x^{6} + 3 x^{5}\right) dx.
Consider the function y = e^{ \sqrt x} and hence find \int \dfrac{4e^{ \sqrt x}}{\sqrt x} \ dx.
Find the following indefinite integrals:
\int 28\left(6x^3 + 1 \right) \left(3x^4 + 2x \right)^6 \ dx
\int \sin3x \cos 3x \ dx
\int 2e^{ \sin x} \cos x \ dx
\int (x + 2) e^{ 3 x^{2} + 12x} \ dx
Consider the function y = \dfrac{\cos x}{e^x}.
Calculate \dfrac{d y}{d x}.
Hence find \int \dfrac{\sin x + \cos x}{e^x} dx.
Consider the function y = e^x \sin x.
Calculate \dfrac{d y}{d x}.
Hence find \int 3e^x(\sin x + \cos x) \ dx.
Consider the function y = \tan x.
Calculate \dfrac{d y}{d x} using \tan x =\dfrac{\sin x}{\cos x}.
Hence find \int \dfrac{5}{\cos^2 x} dx.
Given that \sin 3 t = 3 \sin t - 4 \sin ^{3}\left(t\right)
Make 4 \sin ^{3}\left(t\right) the subject.
Hence find \int 3 \sin ^{3}\left(t\right) dt.
Consider the function y = {x}{e^x}.
Calculate \dfrac{d y}{d x}.
Hence show that xe^x = \int xe^x \ dx + \int e^x \ dx.
Rearrange the expression in part (b) to find \int xe^x \ dx.
Consider the function y = \dfrac{x}{e^x}.
Calculate \dfrac{d y}{d x}.
Hence show that \dfrac{x}{e^x} = \int \dfrac{1}{e^x} \ dx - \int \dfrac{x}{e^x} \ dx.
Rearrange the expression in part (b) to find \int \dfrac{x}{e^x} \ dx.
Consider the function y = x \cos x.
Calculate \dfrac{d y}{d x}.
Use the rearrangement method to find \int x \sin x \ dx.